Question

Suppose that Ayumi rolls a pair of fair six-sided dice. Let A be the event that the first die lands on 2 and B be the event that the second die lands on 2. 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 2? What is P(B) the probability that the second die lands on 2? What is P(A and B) the probability that the first die lands on 2 and the second die lands on 2? Are events A and B independent?

Answer: 1/6 1/6 1/36 independent
I got tired of waiting but if you want the points for the questions have at it

Answer 1

Answer:

$\frac{1}{6} ,\frac{1}{6} ,\frac{1}{36}\,,\,independent$

Step-by-step explanation:

Given: Let A be the event that the first die lands on 2 and B be the event that the second die lands on 2.

To find:

P(A), the probability that the first die lands on 2

P(B), the probability that the second die lands on 2

P(A and B): the probability that the first die lands on 2 and the second die lands on 2

Solution:

Probability refers to chances of occurrence of some event.

Probability = number of favourable outcomes/total number of outcomes

Sample space = $\left \{ 1,2,3,4,5,6 \right \}$

Total number of outcomes = 6

For P(A):

Number of favourable outcomes = 1

So,

$P(A)=\frac{1}{6}$

For P(B):

Number of favourable outcomes = 1

So,

$P(B)=\frac{1}{6}$

P(A and B) = $P(A)P(B)=\left ( \frac{1}{6} \right )\left ( \frac{1}{6} \right )=\frac{1}{36}$

Yes, A and B are independent events as happening of each of the event does not depend on the other.