Probability lab
Explore the tool and practise with 120 lessons in four difficulty levels. Choose a topic, check your answer and take a level test. Your answers stay on this device.
Coin toss and independent trials simulator
For a fair coin leave p = 0.5 and choose the number of tosses. Each click starts a fresh series. A pseudorandom simulation illustrates the model; it does not predict lottery results.
How do I use this page?
- Read a lesson and its worked example.
- Enter an answer, check it and compare the solution steps.
- Generate another exercise or take the 10-question test.
What this page does not do: predict future lottery outcomes.
Course: 120 lessons
- 1. Events and probability
- 2. The complement
- 3. Independent trials
- 4. At least one success
- 5. Unordered combinations
- 6. Expected successes
- 7. Streaks and the gambler’s fallacy
- 8. Disjoint events
- 9. Union of events
- 10. Conditional probability
- 11. Two-stage tree
- 12. Total probability
- 13. Bayes from counts
- 14. Ordered pairs
- 15. Combinations of three
- 16. Number of faces
- 17. Even outcome
- 18. Impossible event
- 19. Certain event
- 20. Disjoint outcomes
- 21. Empirical frequency
- 22. Percentage to probability
- 23. Expected count
- 24. Failed trials
- 25. Frequency error
- 26. Complement of a fraction
- 27. Double complement
- 28. Union count
- 29. Only A
- 30. Outside the union
- 31. Without replacement
- 32. Net expected value
- 33. Variance of successes
- 34. Exactly one success
- 35. Exactly two successes
- 36. At most one success
- 37. Two red draws with replacement
- 38. One red without replacement
- 39. Expected value of a sum
- 40. Scaling payouts
- 41. Two-success path
- 42. Success–failure path
- 43. Total probability
- 44. Three independent stages
- 45. Conditional from joint
- 46. Sum two
- 47. Sum seven
- 48. Doubles
- 49. At least one six
- 50. Different faces
- 51. Code with repetition
- 52. Code without repetition
- 53. Selecting a pair
- 54. Ordering three
- 55. Two selection stages
- 56. First red
- 57. Second after red
- 58. Red then white
- 59. Two whites
- 60. Expected reds
- 61. Variance of independent sums
- 62. Scaling a random variable
- 63. First success
- 64. Mean waiting time
- 65. Simulation frequency error
- 66. Zero successes
- 67. All successes
- 68. One of two
- 69. Binomial mean
- 70. Binomial variance
- 71. True alert count
- 72. False alerts
- 73. Defect given alert
- 74. Defect given no alert
- 75. Classifier precision
- 76. Success on trial two
- 77. More than two trials
- 78. Success by trial three
- 79. Failures before success
- 80. Memorylessness
- 81. Gross payout
- 82. Cost and net result
- 83. Sum of expectations
- 84. Break-even price
- 85. Repeated expectation
- 86. Variance with covariance
- 87. Variance of a difference
- 88. Adding a constant
- 89. Negative scaling
- 90. Mean of independent variables
- 91. Total simulated trials
- 92. Weighting runs
- 93. Monte Carlo standard error
- 94. Reducing simulation error
- 95. Model–sample gap
- 96. Specified streak
- 97. Window count
- 98. Expected matching windows
- 99. Either-side streak
- 100. Success after a streak
- 101. Pair comparisons
- 102. Pair in a day model
- 103. Expected birthday pairs
- 104. Pigeonhole principle
- 105. Three distinct dates
- 106. Different probabilities
- 107. No success with different p
- 108. Perfect dependence
- 109. Unobserved trials
- 110. Distribution total
- 111. Trial budget
- 112. Expected loss
- 113. Break-even probability
- 114. Expected return ratio
- 115. Certain cost
- 116. One in N
- 117. Absolute decrease
- 118. Relative decrease
- 119. Independent next outcome
- 120. Expectation is not a guarantee
Enter a fraction such as 1/3 or round as instructed beside the question (three or eight decimal places).
Optional local storage only. No account or server submission. Turn off to remove saved progress.
1. Events and probability
Level 1
For equally likely outcomes, P(A) is the number of favourable outcomes divided by the total. It lies between 0 and 1.
Worked example
3 red balls out of 10: P = 3/10 = 0.3 = 30%.
Watch out
Do not divide by the number of balls of the other colour. Check that every ball is equally likely.
2. The complement
Level 1
An event and its complement cover all possibilities and cannot occur together. P(not A) = 1 − P(A).
Worked example
A rain probability of 0.2 means a no-rain probability of 0.8 in the same model.
Watch out
The complement is everything outside A, not an arbitrary alternative event.
3. Independent trials
Level 1
Multiply probabilities for independent events. A fair coin has P(heads) = 1/2 on each independent toss.
Worked example
Three heads: 1/2 × 1/2 × 1/2 = 1/8.
Watch out
Do not assume independence. Drawing without replacement changes the pool.
4. At least one success
Level 1
Calculate no successes and subtract from 1. For n independent trials with fixed p: 1 − (1 − p)^n.
Worked example
Two fair tosses: 1 − (1/2)² = 3/4.
Watch out
At least one is not exactly one. Two successes also qualify.
5. Unordered combinations
Level 1
The pair A,B is the same selection as B,A. There are n(n−1)/2 pairs from n distinct items.
Worked example
From 5 items you can select 5 × 4 / 2 = 10 pairs.
Watch out
Do not divide by 2 if order matters, such as selecting a chair and deputy.
6. Expected successes
Level 1
In n trials with success probability p, the expected count is np. It need not be an integer.
Worked example
5 fair tosses average 2.5 heads across repeated series.
Watch out
Expectation is not a promise for one series.
7. Streaks and the gambler’s fallacy
Level 1
Past independent outcomes do not change the next probability. A long streak can look unusual but creates no debt to repay.
Worked example
After five heads, the next independent fair toss still has P(heads) = 1/2.
Watch out
Do not treat history as a guarantee. Simulation illustrates variation, not prediction.
8. Disjoint events
Level 1
For events with no shared outcomes, add their probabilities.
Worked example
On a die: P(1 or 2) = 1/6 + 1/6 = 1/3.
Watch out
Disjoint does not mean independent.
9. Union of events
Level 1
P(A or B) = P(A) + P(B) − P(A and B). Count the overlap once.
Worked example
0.4 + 0.3 − 0.1 = 0.6.
Watch out
Do not add without subtracting the overlap.
10. Conditional probability
Level 1
P(A∣B) is the proportion of A within B; the denominator contains only B.
Worked example
B contains 8 people, 3 also in A: P(A∣B)=3/8.
Watch out
P(A∣B) usually differs from P(B∣A).
11. Two-stage tree
Level 1
A path probability multiplies its successive conditional probabilities.
Worked example
P(A)=0.5, P(B∣A)=0.4: the joint path has probability 0.2.
Watch out
Use the correct branch at the second stage.
12. Total probability
Level 1
Weight the outcome in each disjoint branch and add.
Worked example
Two equally selected bags: probabilities 0.2 and 0.6 give 0.4.
Watch out
An ordinary mean works only for equal weights.
13. Bayes from counts
Level 1
After observing an outcome, compare compatible cases: the source share among all such outcomes.
Worked example
Of 12 red balls, 9 came from A: P(A∣red)=9/12.
Watch out
Do not ignore source frequencies before observation.
14. Ordered pairs
Level 1
With order and no repetition, there are n(n−1) pairs.
Worked example
From 5 items: 5 × 4 = 20.
Watch out
Do not divide by 2 when AB and BA differ.
15. Combinations of three
Level 1
Unordered triples: C(n,3)=n(n−1)(n−2)/6.
Worked example
C(5,3)=5 × 4 × 3 / 6=10.
Watch out
Dividing by 6 removes every ordering of the triple.
16. Number of faces
Level 1 · Elementary outcomes
For equally likely outcomes P(A)=favourable count/total count.
Worked example
Fair 10-sided die. Probability of a specified face? 1/10 = 0.1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
17. Even outcome
Level 1 · Elementary outcomes
For equally likely outcomes P(A)=favourable count/total count.
Worked example
Fair die has faces 1 to 10. Probability of even? 5/10 = 0.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
18. Impossible event
Level 1 · Elementary outcomes
For equally likely outcomes P(A)=favourable count/total count.
Worked example
Standard die 1–6. Probability of 7? 0/6 = 0
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
19. Certain event
Level 1 · Elementary outcomes
For equally likely outcomes P(A)=favourable count/total count.
Worked example
Standard die. Probability of 1 through 6? 6/6 = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
20. Disjoint outcomes
Level 1 · Elementary outcomes
For equally likely outcomes P(A)=favourable count/total count.
Worked example
Fair die 1–6. Probability of 1 or 3 or 5? 3/6 = 0.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
21. Empirical frequency
Level 1 · Proportions and trials
Frequency is a sample result; probability belongs to a model. They are not identical.
Worked example
4 successes in 20 trials. Frequency? 4/20 = 0.2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
22. Percentage to probability
Level 1 · Proportions and trials
Frequency is a sample result; probability belongs to a model. They are not identical.
Worked example
40% as a number from 0 to 1? 40/100 = 0.4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
23. Expected count
Level 1 · Proportions and trials
Frequency is a sample result; probability belongs to a model. They are not identical.
Worked example
p=0.4, n=100. Expected successes? 100×0.4 = 40
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
24. Failed trials
Level 1 · Proportions and trials
Frequency is a sample result; probability belongs to a model. They are not identical.
Worked example
In 20 trials, 4 successes. Failures? 20−4 = 16
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
25. Frequency error
Level 1 · Proportions and trials
Frequency is a sample result; probability belongs to a model. They are not identical.
Worked example
Model p=0.5, observed frequency 0.4. Absolute difference? |0.4−.5| = 0.1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
26. Complement of a fraction
Level 1 · Complements and sets
Complement probability is 1−p. Subtract overlap when adding events.
Worked example
P(A)=4/10. P(Aᶜ)? 1−0.4 = 0.6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
27. Double complement
Level 1 · Complements and sets
Complement probability is 1−p. Subtract overlap when adding events.
Worked example
P(A)=0.4. P((Aᶜ)ᶜ)? (Aᶜ)ᶜ=A = 0.4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
28. Union count
Level 1 · Complements and sets
Complement probability is 1−p. Subtract overlap when adding events.
Worked example
A has 9, B 8, overlap 5. Union count? |A|+|B|−|A∩B| = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
29. Only A
Level 1 · Complements and sets
Complement probability is 1−p. Subtract overlap when adding events.
Worked example
A has 7, overlap B 3. Only A count? |A|−|A∩B| = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
30. Outside the union
Level 1 · Complements and sets
Complement probability is 1−p. Subtract overlap when adding events.
Worked example
100 outcomes, 40 in A∪B. Probability outside union? 1−40/100 = 0.6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
31. Without replacement
Level 2
After drawing a red ball, both the red count and total count decrease by one. Multiply the first probability by the conditional second probability.
Worked example
4 red out of 10: two reds have probability 4/10 × 3/9 = 2/15.
Watch out
Do not reuse 4/10 for the second draw. That would be replacement.
32. Net expected value
Level 2
Multiply each payout by its probability, add the results and subtract the cost. Use one currency throughout.
Worked example
Payout 4 with probability 1/4, cost 2: 4 × 1/4 − 2 = −1.
Watch out
Do not confuse payout with profit. Refunding the ticket price gives zero profit.
33. Variance of successes
Level 2
For independent Bernoulli trials, the variance of the success count is np(1−p). Standard deviation is its square root.
Worked example
8 fair tosses: variance 8 × 1/2 × 1/2 = 2.
Watch out
Variance describes spread, not a fault in the random device.
34. Exactly one success
Level 2
For n independent trials with p=1/2, one success has n possible positions.
Worked example
For 3 tosses: 3 × (1/2)³ = 3/8.
Watch out
At least one and exactly one are different events.
35. Exactly two successes
Level 2
Choose the two success positions and include each sequence probability.
Worked example
In 4 fair tosses: C(4,2)/2⁴ = 6/16.
Watch out
The combination coefficient counts positions, not probability.
36. At most one success
Level 2
Add the disjoint cases: zero and exactly one success.
Worked example
In 3 tosses: (1+3)/8 = 1/2.
Watch out
At most includes zero.
37. Two red draws with replacement
Level 2
Replacing the ball restores the bag; independent draws multiply the same probabilities.
Worked example
3 red out of 10: (3/10)² = 0.09.
Watch out
Without replacement the second numerator and denominator change.
38. One red without replacement
Level 2
Count both orders: red–white and white–red.
Worked example
3 red out of 10: 2 × 3/10 × 7/9 = 7/15.
Watch out
Omitting one order halves the result.
39. Expected value of a sum
Level 2
Expectations add even without independence, when they exist.
Worked example
E(X)=2 and E(Y)=5, so E(X+Y)=7.
Watch out
This rule does not transfer directly to variance.
40. Scaling payouts
Level 2
For constants c,d: E(cX+d)=cE(X)+d.
Worked example
E(X)=3, so E(2X−1)=5.
Watch out
Subtract the cost once per paid trial.
41. Two-success path
Level 2 · Probability trees
Multiply conditional probabilities along a path; add disjoint paths.
Worked example
P(A)=0.4, P(B|A)=0.5. P(A∩B)? 0.4×.5 = 0.2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
42. Success–failure path
Level 2 · Probability trees
Multiply conditional probabilities along a path; add disjoint paths.
Worked example
P(A)=0.4, P(B|A)=0.5. P(A∩Bᶜ)? 0.4×(1−.5) = 0.2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
43. Total probability
Level 2 · Probability trees
Multiply conditional probabilities along a path; add disjoint paths.
Worked example
Bags equally likely. P(red|A)=0.4, P(red|B)=0.2. P(red)? .5×0.4+.5×.2 = 0.3
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
44. Three independent stages
Level 2 · Probability trees
Multiply conditional probabilities along a path; add disjoint paths.
Worked example
Three independent stages, each success probability 0.5. All succeed? .5³ = 0.125
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
45. Conditional from joint
Level 2 · Probability trees
Multiply conditional probabilities along a path; add disjoint paths.
Worked example
P(A∩B)=0.2, P(A)=0.4. P(B|A)? P(A∩B)/P(A) = 0.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
46. Sum two
Level 2 · Two dice
Two independent fair dice give 36 equally likely ordered pairs. Their sums are not equally likely.
Worked example
Probability of sum 2? 1/36 = 0.02777777778
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
47. Sum seven
Level 2 · Two dice
Two independent fair dice give 36 equally likely ordered pairs. Their sums are not equally likely.
Worked example
Probability of sum 7? 6/36 = 0.1666666667
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
48. Doubles
Level 2 · Two dice
Two independent fair dice give 36 equally likely ordered pairs. Their sums are not equally likely.
Worked example
Probability of matching faces? 6/36 = 0.1666666667
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
49. At least one six
Level 2 · Two dice
Two independent fair dice give 36 equally likely ordered pairs. Their sums are not equally likely.
Worked example
Probability of at least one six? 1−(5/6)² = 0.3055555556
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
50. Different faces
Level 2 · Two dice
Two independent fair dice give 36 equally likely ordered pairs. Their sums are not equally likely.
Worked example
Probability of different faces? 1−6/36 = 0.8333333333
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
51. Code with repetition
Level 2 · Counting configurations
Distinguish order and repetition. The product rule needs the correct number of choices at each stage.
Worked example
Three digits 0–9, repeats allowed. How many codes? 10³ = 1000
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
52. Code without repetition
Level 2 · Counting configurations
Distinguish order and repetition. The product rule needs the correct number of choices at each stage.
Worked example
Three distinct digits 0–9, leading zero allowed. Count? 10×9×8 = 720
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
53. Selecting a pair
Level 2 · Counting configurations
Distinguish order and repetition. The product rule needs the correct number of choices at each stage.
Worked example
Unordered pairs from 5 people. Count? n(n−1)/2 = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
54. Ordering three
Level 2 · Counting configurations
Distinguish order and repetition. The product rule needs the correct number of choices at each stage.
Worked example
How many orders of 3 distinct people? 3!=6 = 6
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
55. Two selection stages
Level 2 · Counting configurations
Distinguish order and repetition. The product rule needs the correct number of choices at each stage.
Worked example
4 shirts and 3 trousers. Outfits? 4×3 = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
56. First red
Level 2 · Sampling without replacement
Without replacement, counts and denominators change. Do not assume independence.
Worked example
10 balls, 4 red. P(first red)? 4/10 = 0.4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
57. Second after red
Level 2 · Sampling without replacement
Without replacement, counts and denominators change. Do not assume independence.
Worked example
10 balls, 4 red. First red removed. P(second red)? 3/9 = 0.3333333333
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
58. Red then white
Level 2 · Sampling without replacement
Without replacement, counts and denominators change. Do not assume independence.
Worked example
10 balls, 4 red. Without replacement: P(red then white)? 4/10×6/9 = 0.2666666667
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
59. Two whites
Level 2 · Sampling without replacement
Without replacement, counts and denominators change. Do not assume independence.
Worked example
10 balls, 4 red. Without replacement: P(two whites)? 6/10×5/9 = 0.3333333333
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
60. Expected reds
Level 2 · Sampling without replacement
Without replacement, counts and denominators change. Do not assume independence.
Worked example
10 balls, 4 red. Draw 2 without replacement. Expected reds? 2×4/10 = 0.8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
61. Variance of independent sums
Level 3
Independent variances add. Standard deviations do not.
Worked example
Variances 2 and 3 give sum variance 5.
Watch out
Dependence requires a covariance term.
62. Scaling a random variable
Level 3
Var(cX)=c²Var(X). Adding a constant does not change variance.
Worked example
Var(X)=2: Var(3X)=18.
Watch out
Square the multiplier.
63. First success
Level 3
First success on trial k requires k−1 failures followed by a success.
Worked example
For p=1/2 on trial 3: (1/2)³=1/8.
Watch out
Do not multiply by positions: success is last.
64. Mean waiting time
Level 3
For independent trials with fixed p>0, expected trials until success equal 1/p.
Worked example
p=1/5 gives 5 trials on average, with no guaranteed deadline.
Watch out
The mean does not promise success by that trial.
65. Simulation frequency error
Level 3
Compare observed frequency with p. Multiply the difference by 100 for percentage points.
Worked example
60/100 versus p=0.5: absolute difference 10 pp.
Watch out
Percentage points are not relative percentage change.
66. Zero successes
Level 3 · Binomial model
The model requires independent trials with fixed p. P(X=k)=C(n,k)p^k(1−p)^(n−k).
Worked example
n=5, p=0.5. P(X=0)? (.5)^5 = 0.03125
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
67. All successes
Level 3 · Binomial model
The model requires independent trials with fixed p. P(X=k)=C(n,k)p^k(1−p)^(n−k).
Worked example
n=3, p=0.4. P(X=3)? 0.4³ = 0.064
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
68. One of two
Level 3 · Binomial model
The model requires independent trials with fixed p. P(X=k)=C(n,k)p^k(1−p)^(n−k).
Worked example
n=2, p=0.4. P(X=1)? 2p(1−p) = 0.48
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
69. Binomial mean
Level 3 · Binomial model
The model requires independent trials with fixed p. P(X=k)=C(n,k)p^k(1−p)^(n−k).
Worked example
n=5, p=0.4. E(X)? 5×0.4 = 2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
70. Binomial variance
Level 3 · Binomial model
The model requires independent trials with fixed p. P(X=k)=C(n,k)p^k(1−p)^(n−k).
Worked example
n=5, p=0.4. Var(X)? np(1−p) = 1.2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
71. True alert count
Level 3 · Bayes and natural frequencies
Convert groups to counts. P(A|B) generally differs from P(B|A).
Worked example
1,000 packages, 10% defective. Detector finds 80% of defects. True alerts? 1000×.1×.8 = 80
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
72. False alerts
Level 3 · Bayes and natural frequencies
Convert groups to counts. P(A|B) generally differs from P(B|A).
Worked example
900 good packages, 10% trigger an alert. False alerts? 900×.1 = 90
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
73. Defect given alert
Level 3 · Bayes and natural frequencies
Convert groups to counts. P(A|B) generally differs from P(B|A).
Worked example
80 true and 90 false alerts. P(defect|alert)? 80/(80+90) = 0.4705882353
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
74. Defect given no alert
Level 3 · Bayes and natural frequencies
Convert groups to counts. P(A|B) generally differs from P(B|A).
Worked example
No alert: 20 defective, 810 good. P(defect|no alert)? 20/(20+810) = 0.02409638554
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
75. Classifier precision
Level 3 · Bayes and natural frequencies
Convert groups to counts. P(A|B) generally differs from P(B|A).
Worked example
4 true alerts and 3 false alerts. Precision? true/(true+false) = 0.5714285714
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
76. Success on trial two
Level 3 · Waiting for success
For independent trials with fixed p, waiting time includes the successful trial: E(T)=1/p.
Worked example
p=0.4. P(T=2)? (1−p)p = 0.24
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
77. More than two trials
Level 3 · Waiting for success
For independent trials with fixed p, waiting time includes the successful trial: E(T)=1/p.
Worked example
p=0.4. P(T>2)? (1−p)² = 0.36
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
78. Success by trial three
Level 3 · Waiting for success
For independent trials with fixed p, waiting time includes the successful trial: E(T)=1/p.
Worked example
p=0.4. P(T≤3)? 1−(1−p)³ = 0.784
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
79. Failures before success
Level 3 · Waiting for success
For independent trials with fixed p, waiting time includes the successful trial: E(T)=1/p.
Worked example
p=1/5. Expected failures before first success? (1−p)/p = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
80. Memorylessness
Level 3 · Waiting for success
For independent trials with fixed p, waiting time includes the successful trial: E(T)=1/p.
Worked example
Fixed p=0.4, already 5 failures. Success chance on the next independent trial? P(next success)=p = 0.4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
81. Gross payout
Level 3 · Expected value
Expectation is the sum of values times probabilities. It is not a promise for a single trial.
Worked example
Payout 30 with probability 0.4, otherwise 0. Expected payout? 30×0.4 = 12
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
82. Cost and net result
Level 3 · Expected value
Expectation is the sum of values times probabilities. It is not a promise for a single trial.
Worked example
Expected payout=4, cost=3. Expected net? E(payout)−cost = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
83. Sum of expectations
Level 3 · Expected value
Expectation is the sum of values times probabilities. It is not a promise for a single trial.
Worked example
E(X)=4, E(Y)=3. E(2X+Y)? 2E(X)+E(Y) = 11
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
84. Break-even price
Level 3 · Expected value
Expectation is the sum of values times probabilities. It is not a promise for a single trial.
Worked example
Payout 20 with probability 0.25, otherwise 0. Price giving expected net zero? 20×.25 = 5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
85. Repeated expectation
Level 3 · Expected value
Expectation is the sum of values times probabilities. It is not a promise for a single trial.
Worked example
Expected net per trial = −0.2. Expected net for 100 trials? 100×(−.2) = -20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
86. Variance with covariance
Level 3 · Dependence and variance
Var(X+Y)=Var(X)+Var(Y)+2cov(X,Y). Independence implies zero covariance, but not always conversely.
Worked example
Var(X)=4, Var(Y)=9, cov=2. Var(X+Y)? 4+9+2×2 = 17
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
87. Variance of a difference
Level 3 · Dependence and variance
Var(X+Y)=Var(X)+Var(Y)+2cov(X,Y). Independence implies zero covariance, but not always conversely.
Worked example
Var(X)=4, Var(Y)=9, cov=2. Var(X−Y)? 4+9−2×2 = 9
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
88. Adding a constant
Level 3 · Dependence and variance
Var(X+Y)=Var(X)+Var(Y)+2cov(X,Y). Independence implies zero covariance, but not always conversely.
Worked example
Var(X)=4. Var(X+3)? Var(X+c)=Var(X) = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
89. Negative scaling
Level 3 · Dependence and variance
Var(X+Y)=Var(X)+Var(Y)+2cov(X,Y). Independence implies zero covariance, but not always conversely.
Worked example
Var(X)=4. Var(−2X)? (−2)²Var(X) = 16
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
90. Mean of independent variables
Level 3 · Dependence and variance
Var(X+Y)=Var(X)+Var(Y)+2cov(X,Y). Independence implies zero covariance, but not always conversely.
Worked example
Independent X,Y each have variance 8. Var((X+Y)/2)? (2a+2a)/4 = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
91. Total simulated trials
Level 4 · Project: simulation
Compare a model with many runs. One run does not validate a generator or forecast.
Worked example
4 runs of 30 trials. Total? 4×30 = 120
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
92. Weighting runs
Level 4 · Project: simulation
Compare a model with many runs. One run does not validate a generator or forecast.
Worked example
Run A: 1/2 successes, B: 8/10. Combined frequency? (1+8)/(2+10) = 0.75
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
93. Monte Carlo standard error
Level 4 · Project: simulation
Compare a model with many runs. One run does not validate a generator or forecast.
Worked example
p=0.5, n=100. SD of simulated frequency? √(.5×.5/100) = 0.05
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
94. Reducing simulation error
Level 4 · Project: simulation
Compare a model with many runs. One run does not validate a generator or forecast.
Worked example
To halve SE, by what factor must independent trial count increase? SE ∝ 1/√n = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
95. Model–sample gap
Level 4 · Project: simulation
Compare a model with many runs. One run does not validate a generator or forecast.
Worked example
Model expects 50, observed 54. Count deviation? observed−expected = 4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
96. Specified streak
Level 4 · Project: streaks
A specified streak differs from a streak anywhere. Overlapping windows are dependent.
Worked example
5 specified fair tosses must all be heads. Probability? 2^(−n) = 0.03125
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
97. Window count
Level 4 · Project: streaks
A specified streak differs from a streak anywhere. Overlapping windows are dependent.
Worked example
10 tosses, windows of length 3. Possible starts? 10−3+1 = 8
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
98. Expected matching windows
Level 4 · Project: streaks
A specified streak differs from a streak anywhere. Overlapping windows are dependent.
Worked example
10 fair tosses. Expected HHH windows? 8×1/8 = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
99. Either-side streak
Level 4 · Project: streaks
A specified streak differs from a streak anywhere. Overlapping windows are dependent.
Worked example
First 3 fair tosses identical (HHH or TTT). Probability? 2/8 = 0.25
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
100. Success after a streak
Level 4 · Project: streaks
A specified streak differs from a streak anywhere. Overlapping windows are dependent.
Worked example
After 10 failures, next 2 independent trials have p=0.5. Chance of at least one success? 1−.5² = 0.75
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
101. Pair comparisons
Level 4 · Project: collisions
Compare any pair; there are n(n−1)/2 pairs. Expected pair count is not collision probability.
Worked example
5 people. Pair count? n(n−1)/2 = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
102. Pair in a day model
Level 4 · Project: collisions
Compare any pair; there are n(n−1)/2 pairs. Expected pair count is not collision probability.
Worked example
Two people, 365 independent equally likely days. Matching probability? 1/365 = 0.002739726027
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
103. Expected birthday pairs
Level 4 · Project: collisions
Compare any pair; there are n(n−1)/2 pairs. Expected pair count is not collision probability.
Worked example
10 people, 365 days. Expected matching pairs? C(10,2)/365 = 0.1232876712
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
104. Pigeonhole principle
Level 4 · Project: collisions
Compare any pair; there are n(n−1)/2 pairs. Expected pair count is not collision probability.
Worked example
366 people and only 365 possible days. Probability of at least one collision? n>d → certain = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
105. Three distinct dates
Level 4 · Project: collisions
Compare any pair; there are n(n−1)/2 pairs. Expected pair count is not collision probability.
Worked example
3 people, 10 equally likely independent days. Probability all dates differ? 1×9/10×8/10 = 0.72
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
106. Different probabilities
Level 4 · Project: model validation
Check independence, constant p and complete data before applying formulas.
Worked example
Two independent trials p1=0.2, p2=0.8. Expected successes? .2+.8 = 1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
107. No success with different p
Level 4 · Project: model validation
Check independence, constant p and complete data before applying formulas.
Worked example
Independent p1=0.2, p2=0.8. P(zero successes)? .8×.2 = 0.16
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
108. Perfect dependence
Level 4 · Project: model validation
Check independence, constant p and complete data before applying formulas.
Worked example
Second outcome copies the first fair toss exactly. P(two heads)? both heads iff first is heads = 0.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
109. Unobserved trials
Level 4 · Project: model validation
Check independence, constant p and complete data before applying formulas.
Worked example
100 trials, only 80 outcomes known. Unknown percentage? 20/100×100 = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
110. Distribution total
Level 4 · Project: model validation
Check independence, constant p and complete data before applying formulas.
Worked example
Distribution probabilities 0.2; 0.3; x. Find x. 1−.2−.3 = 0.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
111. Trial budget
Level 4 · Project: decisions and cost
Separate payout from profit and risk from expectation. Calculations are not betting advice.
Worked example
Budget 40, cost per trial 4. Trial count? budget/cost = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
112. Expected loss
Level 4 · Project: decisions and cost
Separate payout from profit and risk from expectation. Calculations are not betting advice.
Worked example
Cost 2, expected payout 1.5. Expected loss over 20 trials? 20×(2−1.5) = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
113. Break-even probability
Level 4 · Project: decisions and cost
Separate payout from profit and risk from expectation. Calculations are not betting advice.
Worked example
Cost 1, payout 5 or 0. Which p gives expected net zero? 5p−1=0 = 0.2
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
114. Expected return ratio
Level 4 · Project: decisions and cost
Separate payout from profit and risk from expectation. Calculations are not betting advice.
Worked example
Cost 10, expected payout 4. Expected payout/cost ratio? 4/10 = 0.4
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
115. Certain cost
Level 4 · Project: decisions and cost
Separate payout from profit and risk from expectation. Calculations are not betting advice.
Worked example
5 trials costing 3 each. Total cost regardless of outcomes? 5×3 = 15
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
116. One in N
Level 4 · Project: communicating uncertainty
State denominator, horizon and assumptions. Relative change is not a percentage-point difference.
Worked example
P=1/5. Percentage? 100/5 = 20
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
117. Absolute decrease
Level 4 · Project: communicating uncertainty
State denominator, horizon and assumptions. Relative change is not a percentage-point difference.
Worked example
P falls from 0.2 to 0.1. Drop in percentage points? (.2−.1)×100 = 10
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
118. Relative decrease
Level 4 · Project: communicating uncertainty
State denominator, horizon and assumptions. Relative change is not a percentage-point difference.
Worked example
P falls from 0.2 to 0.1. Relative decrease percent? (.2−.1)/.2×100 = 50
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
119. Independent next outcome
Level 4 · Project: communicating uncertainty
State denominator, horizon and assumptions. Relative change is not a percentage-point difference.
Worked example
100 past draws. Next independent event probability is 0.1. Give its probability after seeing history. independence preserves p = 0.1
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
120. Expectation is not a guarantee
Level 4 · Project: communicating uncertainty
State denominator, horizon and assumptions. Relative change is not a percentage-point difference.
Worked example
Two fair coins: expected heads=1. Probability of exactly one head? 2/4 = 0.5
Watch out
State the requested unit. Enter a fraction or round non-terminating results to eight decimal places. Check the assumptions stated in this lesson.
Test: choose your scope
Choose a lesson range and test length. Topics are drawn without repetition, up to the number available in that range. Answers appear after the test; there is no time limit.
Further reading
Original exercises by Pewniaki. The mental-maths identities are explained algebraically; no claim of historical origin or guaranteed speed advantage is made.