Question

You roll a pair of fair six-sided dice. The sample space of 36 possible outcomes is shown below. Based on this information, answer the following questions.
What is P(A)P(A)P, left parenthesis, A, right parenthesis, the probability that the first die is a 3?

What is P(B)P(B)P, left parenthesis, B, right parenthesis, the probability that the sum of the dice is 8?

What is P(A and B)P(A and B)P, left parenthesis, A, start text, space, a, n, d, space, end text, B, right parenthesis, the probability that the first die is a 3 and the sum of the dice is 8?

What is P(B | A)P(B | A)P, left parenthesis, B, start text, space, vertical bar, space, end text, A, right parenthesis, the conditional probability that the sum of the dice is 8 given that the first die is a 3?

Is P(B | A)=P(B)P(B | A)=P(B)P, left parenthesis, B, start text, space, vertical bar, space, end text, A, right parenthesis, equals, P, left parenthesis, B, right parenthesis? Are the events AAA and BBB independent?

Answer 1

Answer:

P(A) = 1/6

P(B) = 5/36

P(A and B) = 1/36

P(B | A) = 1/6

P(B) is different of P(B | A). As they are different, we have that A and B are DEPENDENT events.

Step-by-step explanation:

As one die has six numbers, the probability of the first die being a 3 is 1/6, because there is one number 3 in the six numbers of the dice:

P(A) = 1/6

The dice can have a sum of 8 in the following pair of values:

(2,6), (3,5), (4,4), (5,3), (6,2)

So there are 5 possibilities among the total 36, so P(B) = 5/36

The probability of the first die being 3 and the sum being 8 only happens in one pair of values:

(3,5)

So the probability P(A and B) is 1/36

The probability of the sum being 8 given that the first number is 3 is the probability of having a 5 in the second die, so it is one possibility among 6:

P(B | A) = 1/6

P(B) = 5/36 and P(B | A) = 1/6, so we have that P(B) is different of P(B | A). As they are different, we have that A and B are DEPENDENT events.