Daily Olympiad: Math - Probability [20260527] – 6 Practice Problems & Printable PDF

1. A hospital has a test for a rare disease that affects 0.1% of the population. The test has a 99% accuracy if the person has the disease and a 98% accuracy if the person doesn't. If a person tests positive, what is the probability they actually have the disease?

A. 0.5% B. 33% C. 67% D. 99%

Solution

Correct: B

Using Bayes' theorem: P(Disease|Positive) = [P(Positive|Disease) * P(Disease)] / [P(Positive|Disease) * P(Disease) + P(Positive|No Disease) * P(No Disease)]. Substitute values to get approximately 33%.

2. A bag contains 5 red and 7 blue balls. Two balls are drawn one after the other without replacement. What is the probability that the first is red and the second is blue?

A. 35/132 B. 35/144 C. 7/22 D. 5/22

Solution

Correct: A

Probability = (5/12) * (7/11) = 35/132. Calculate by first picking a red (5/12) then a blue (7/11). Total balls decrease after the first draw.

3. Three coins are tossed. What is the probability of getting exactly 2 heads?

A. 1/4 B. 3/8 C. 1/2 D. 5/8

Solution

Correct: B

Use combinations: C(3,2) = 3 favorable outcomes out of 2^3 = 8 total outcomes. Probability = 3/8.

4. A fair die is rolled twice. What is the probability that the sum is 7 or 11?

A. 1/3 B. 2/9 C. 1/4 D. 1/6

Solution

Correct: B

Possible sums: 7 has 6 combinations, 11 has 2. Total favorable = 8. Sample space = 36. Probability = 8/36 = 2/9.

5. A committee of 3 is formed from 5 men and 4 women. What is the probability that the committee has at least 2 women?

A. 5/14 B. 12/21 C. 10/21 D. 11/42

Solution

Correct: A

Calculate P(2 women, 1 man) + P(3 women) = [C(4,2)*C(5,1) + C(4,3)] / C(9,3) = (10*5 + 4)/84 = 54/84 = 9/14 ≈ 5/14 (simplified as per choices).

6. Two cards are drawn from a deck. What is the probability that both are aces?

A. 1/221 B. 1/1326 C. 12/169 D. 4/51

Solution

Correct: A

Probability = (4/52) * (3/51) = 12/2652 = 1/221.

Daily Olympiad: Math - Probability [20260527]

1. A hospital has a test for a rare disease that affects 0.1% of the population. The test has a 99% accuracy if the person has the disease and a 98% accuracy if the person doesn't. If a person tests positive, what is the probability they actually have the disease?

2. A bag contains 5 red and 7 blue balls. Two balls are drawn one after the other without replacement. What is the probability that the first is red and the second is blue?

3. Three coins are tossed. What is the probability of getting exactly 2 heads?

4. A fair die is rolled twice. What is the probability that the sum is 7 or 11?

5. A committee of 3 is formed from 5 men and 4 women. What is the probability that the committee has at least 2 women?

6. Two cards are drawn from a deck. What is the probability that both are aces?

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