Question

Die A is numbered 1, 2, 3, 4, 5, and 6. Die B is numbered 2, 2, 4, 4, 6, and 6. A fair coin is flipped. If it is heads, die A is rolled; otherwise, die B is rolled. Given the die roll was even, find the probability the coin flip was heads.

a. 1/6
b. 1/3
c. 1/2
d. 2/3

Answer 1

To find the probability that the coin flip was heads given an even die roll, conditional probability and Bayes' theorem are used. The result is a probability of 1/3, indicating the coin flip was heads. Correct option is option b. 1/3

Step-by-step explanation:

The student has asked for help to find the probability that the coin flip was heads, given that the rolled die showed an even number when a fair coin is flipped to decide which of two dice (Die A or Die B) to roll. To find this probability, we use conditional probability and Bayes' theorem.

First, we calculate the probability of rolling an even number on each die:

• Probability of rolling an even number on Die A: P(even | A) = 3/6 = 1/2
• Probability of rolling an even number on Die B: P(even | B) = 3/3 = 1

Next, the probability of rolling die A or B given a coin flip is 1/2, as the coin is fair.

Now, we determine the total probability of rolling an even number:

P(even) = (P(A) × P(even | A)) + (P(B) × P(even | B))

P(even) = (1/2 × 1/2) + (1/2 × 1) = 1/4 + 1/2 = 3/4

The probability that the coin flip was heads given the die roll was even is P(A | even), so we apply Bayes' theorem:

P(A | even) = (P(even | A) × P(A)) / P(even)

P(A | even) = (1/2 × 1/2) / (3/4)

P(A | even) = 1/4 / 3/4

P(A | even) = 1/3

Therefore, the probability that the coin flip was heads given the die roll was even is 1/3, which corresponds to option b.