Probability — Practice

Question 1

A pair of dice is thrown. Find the probability the sum is 7 given the first die shows 3.

Accepted Answer

With the first die fixed at 3, the only way to get sum 7 is (3,4). So P = 1/6.

Question 2

A coin is tossed 5 times. Find the probability of at least 3 heads.

Accepted Answer

n = 5, p = 1/2. P(X >= 3) = (5C3 + 5C4 + 5C5)/32 = (10 + 5 + 1)/32 = 16/32 = 1/2.

Question 3

X takes values 0,1,2,3 with probabilities 0.1, 0.3, 0.4, k. Find k, E(X), and Var(X).

Accepted Answer

Sum = 1 gives k = 0.2. E(X) = 0.3 + 0.8 + 0.6 = 1.7. E(X^2) = 0.3 + 1.6 + 1.8 = 3.7. Var(X) = 3.7 - 1.7^2 = 0.81.

Question 4

A bag has 5 white and 3 black balls. Two are drawn. Find the probability they are different colours.

Accepted Answer

P = (5/8)(3/7) + (3/8)(5/7) = 15/56 + 15/56 = 30/56 = 15/28.

Question 5

An examinee guesses, copies, or knows an answer with probabilities 0.1, 0.2, 0.7; correct with probability 0.2, 0.8, 1 respectively. Find P(knew | correct).

Accepted Answer

P(correct) = 0.1(0.2) + 0.2(0.8) + 0.7(1) = 0.88. P(knew | correct) = 0.7/0.88 = 35/44.

Practice — Probability

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