ISAT - International Student Admissions Probability and Statistical Analysis 2 — Questions and Answers
Question 1: Bag: 5 red, 3 blue, 2 green. Two drawn without replacement. Probability both are red?
- 2/9 (Correct answer)
- 1/4
- 5/18
- 1/5
Correct answer: 2/9
P = (5/10)(4/9) = 20/90 = 2/9.
P(first red) = 5/10. P(second red|first red) = 4/9. P(both) = 20/90 = 2/9.
Question 2: 500 students, heights normally distributed: mean=170cm, SD=8cm. How many are taller than 186cm?
- About 12-13 (Correct answer)
- About 25
- About 1-2
- About 50
Correct answer: About 12-13
186 = mean + 2SD. About 2.5% above +2SD. 0.025 x 500 = 12.5.
z = (186-170)/8 = 2. About 2.5% above +2SD. 2.5% of 500 = 12.5 students.
Question 3: Probability of drawing a King OR a Heart from a standard 52-card deck?
- 4/13 (Correct answer)
- 17/52
- 16/52
- 1/4
Correct answer: 4/13
P = 4/52 + 13/52 - 1/52 = 16/52 = 4/13.
P(King or Heart) = P(King) + P(Heart) - P(King and Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13.
Question 4: A test: 95% sensitivity, 90% specificity. Disease prevalence: 1%. What is the probability a positive result is a true positive?
- Approximately 8.8% (Correct answer)
- 95%
- 90%
- About 50%
Correct answer: Approximately 8.8%
Bayes: (0.95 x 0.01) / (0.95 x 0.01 + 0.10 x 0.99) = 0.0095/0.1085 = 8.8%.
In 10,000 people: 95 true positives, 990 false positives. P(disease|positive) = 95/1085 = 8.8%.
Question 5: Dataset: 12, 15, 18, 18, 20, 22, 25, 28, 30, 35. Calculate the interquartile range.
- 13
- 10 (Correct answer)
- 15
- 12
Correct answer: 10
Q1 = median of lower half = 18. Q3 = median of upper half = 28. IQR = 28 - 18 = 10.
Lower half: 12,15,18,18,20. Q1=18. Upper half: 22,25,28,30,35. Q3=28. IQR=10.
Question 6: A fair die rolled three times. Probability of exactly two 6s?
- 5/72 (Correct answer)
- 1/36
- 25/216
- 15/216
Correct answer: 5/72
P = C(3,2) x (1/6)^2 x (5/6) = 3 x 1/36 x 5/6 = 15/216 = 5/72.
Binomial: C(3,2) x (1/6)^2 x (5/6)^1 = 3 x 1/36 x 5/6 = 15/216 = 5/72.
ISAT - International Student Admissions Probability and Statistical Analysis 2
Bag: 5 red, 3 blue, 2 green.
Two drawn without replacement.
Probability both are red?