Question

Suppose that Antonía rolls a pair of fair six-sided dice. Let A be the event that the first die lands on 5 and

B be the event that the sum of the two dice is 6. Using the sample space of possible outcomes below,
answer each of the following questions.
What is P(A), the probability that the first die lands on 5?
What is P(B), the probability that the sum of the two dice is 6?
What is P A and B), the probability that the first die lands on 5 and the two dice add up to 6?
Are events A and B independent?
Choose 1 answer:
A
Yes, events A and B are independent events.
®
No, events A and B are not independent events.
OD DO SORO DO O​

Answer 1

`P(A)=(1)/(6)`

`P(B)=(5)/(36)`

`P(A\cap B)=(1)/(36)`

No, events A and B are not independent events.

Explanation:

Suppose that Antonía rolls a pair of fair six-sided dice.

All possible outcomes are:

`\begin{array}{cccccc}(1,1)&(1,2)&(1,3)&(1,4)&(1,5)&(1,6)\\(2,1)&(2,2)&(2,3)&(2,4)&(2,5)&(2,6)\\(3,1)&(3,2)&(3,3)&(3,4)&(3,5)&(3,6)\\(4,1)&(4,2)&(4,3)&(4,4)&(4,5)&(4,6)\\(5,1)&(5,2)&(5,3)&(5,4)&(5,5)&(5,6)\\(6,1)&(6,2)&(6,3)&(6,4)&(6,5)&(6,6)\end{array}`

(36 outcomes in total)

A = the first die lands on 5

Favorable outcomes:

`(5,1),\ (5,2),\ (5,3),\ (5,4),\ (5,5),\ (5,6)`

(6 outcomes)

Hence,

`P(A)=(6)/(36)=(1)/(6)`

B = the sum of the two dice is 6

Favorable outcomes:

`(1,5),\ (2,4),\ (3,3),\ (4,2),\ (5,1)`

(5 outcomes)

Hence,

`P(B)=(5)/(36)`

A∩B = the first die lands on 5 and the two dice add up to 6

Favorable outcomes:

`(5,1)`

(1 outcome)

Hence,

`P(A\cap B)=(1)/(36)`

Two events are independent when

`P(A\cap B)=P(A)\cdot P(B)`

Since

`(1)/(36)\\eq (5)/(36)\cdot (1)/(6),`

events are not independent.