Question

You roll two fair six-sided dice, and label the outcomes as d₁ and d₂ Let A and B denote the events associated with the following conditions:

a) Event A: The sum of the two dice is 7.
Event B: The outcome of d₁ is greater than the outcome ofd₂
b) Event A: The sum of the two dice is 9.
Event B: The outcome of d₁ is less than or equal to the outcome of d₂
c) Event A: The sum of the two dice is 5.
Event B: The outcome of d₁is odd.

d) Event A: The sum of the two dice is 12.
Event B: The outcome of d₁is a prime number.

e) Event A: The sum of the two dice is 6.
Event B: The outcome of d₁ and d₂ is the same.

Answer 1

In this question, Event A is defined as either a three or four is rolled first, followed by an even number. To find P(A), you need to determine the number of outcomes in A and divide it by the total number of outcomes in the sample space. Event B is defined as the sum of the two rolls is at most seven. To find P(B), you need to determine the number of outcomes that satisfy B and divide it by the total number of outcomes in the sample space. P(A/B) represents the probability of event A occurring given that event B has already occurred. P(A|B) can be found by multiplying the probabilities of A and B, given that A and B are independent events. A and B are not mutually exclusive events because there are outcomes that satisfy both A and B. A and B are not independent events because P(A AND B) is not equal to P(A) × P(B).

Step-by-step explanation:

b. Let A be the event that either a three or four is rolled first, followed by an even number. The possible outcomes for A are {3, 4, 6, 8, 10, 12}. The sample space for the two dice is {1, 2, 3, 4, 5, 6} × {1, 2, 3, 4, 5, 6}. The total number of outcomes in the sample space is 6 × 6 = 36. Therefore, P(A) = number of outcomes in A / total number of outcomes = 6 / 36 = 1 / 6.

c. Let B be the event that the sum of the two rolls is at most seven. To find P(B), we need to determine the number of outcomes that satisfy B and divide it by the total number of outcomes in the sample space. The possible outcomes for B are {1, 2, 3, 4, 5, 6, 7}. There are 21 outcomes that satisfy B out of a total of 36 outcomes in the sample space. Therefore, P(B) = 21 / 36 = 7 / 12.

d. P(A/B) represents the probability of event A occurring given that event B has already occurred. To find P(A/B), we need to determine the number of outcomes that satisfy both A and B and divide it by the number of outcomes that satisfy B. Since A and B are independent events, the number of outcomes that satisfy both A and B is the product of the probabilities of A and B. P(A) = 1 / 6, and P(B) = 7 / 12. Therefore, P(A/B) = P(A) × P(B) = (1 / 6) × (7 / 12) = 7 / 72.

e. A and B are not mutually exclusive events because there are outcomes that satisfy both A and B. The outcomes {4, 6} satisfy both A and B. Numerically, P(A AND B) = number of outcomes in (A AND B) / total number of outcomes = 2 / 36 = 1 / 18. Since P(A AND B) is not zero, A and B are not mutually exclusive.

f. A and B are independent events if the occurrence of one event does not affect the probability of the other event. To determine if A and B are independent, we need to compare P(A) × P(B) with P(A AND B). P(A) = 1 / 6, P(B) = 7 / 12, and P(A AND B) = 1 / 18. P(A) × P(B) = (1 / 6) × (7 / 12) = 7 / 72. Since P(A AND B) is not equal to P(A) × P(B), A and B are not independent events.