Question

A pair of dice consisting of a six-sided die and a four-sided die is rolled

and the sum is determined. Let A be the event that a sum of 5 is rolled and
let B be the event that a sum of 5 or a sum of 9 is rolled. Find (a) P(A), (b)
P(B), and (c) P(A \ B).

Answer 1

To find probabilities of events A and B, we consider the total number of outcomes and the number of favorable outcomes for each event. P(A) is 1/6, P(B) is 1/3, and P(A|B) is 1/2.

Step-by-step explanation:

To find the probabilities of the events mentioned, we need to determine the total number of outcomes and the number of favorable outcomes for each event.

(a) P(A) = Number of favorable outcomes for event A / Total number of outcomes

In this case, event A is getting a sum of 5, which can be obtained in 4 ways - (1, 4), (2, 3), (3, 2), and (4, 1). The total number of outcomes is the product of the number of sides on each die, which is 6 * 4 = 24. Therefore, P(A) = 4/24 = 1/6.

(b) P(B) = Number of favorable outcomes for event B / Total number of outcomes

Event B is getting a sum of 5 or a sum of 9. The favorable outcomes for getting a sum of 5 are the same as in event A (4 outcomes), and the favorable outcomes for getting a sum of 9 are (3, 6), (4, 5), (5, 4), and (6, 3) (4 outcomes). The total number of outcomes is still 24. Therefore, P(B) = (4 + 4)/24 = 8/24 = 1/3.

(c) P(A ∩ B) = Number of favorable outcomes for event A ∩ B / Total number of outcomes

Event A ∩ B is getting a sum of 5 and a sum of 9, which corresponds to the outcomes (1, 4), (2, 3), (3, 2), and (4, 1). The favorable outcomes are the same as event A, which is 4. Therefore, P(A ∩ B) = 4/24 = 1/6.

P(A|B) = P(A ∩ B) / P(B)

Substituting the values we've calculated, P(A|B) = (1/6) / (1/3) = 1/2.