Question

From the sample space S={1,2,3,4,5,6,7,8}, a single number is selected randomly. Given the events
A: selected number is even
B: selected number is a multiple of 3
Find each probability.

P(A)=

P(B)=

P(A and B)=

P(B ┤| A)=

P(A ┤| B)=

Two events A and B are called independent events if knowledge about the occurrence of one of them has no effect on the probability of the other one, that is, if
P(B ┤| A)=P(B), or equivalently P(A ┤| B)=P(A).
Are the events A and B independent? Explain why or why not.

Answer 1

To find each probability, we need to count how many numbers satisfy the condition and divide by the total number of elements in the sample space.

1) P(A): The selected number is even.
There are 4 even numbers in the sample space: 2, 4, 6, and 8.
P(A) = Number of favorable outcomes / Total number of outcomes = 4 / 8 = 1/2

2) P(B): The selected number is a multiple of 3.
There are 2 numbers in the sample space that are multiples of 3: 3 and 6.
P(B) = Number of favorable outcomes / Total number of outcomes = 2 / 8 = 1/4

3) P(A and B): The selected number is both even and a multiple of 3.
There is only 1 number in the sample space that satisfies this condition: 6.
P(A and B) = Number of favorable outcomes / Total number of outcomes = 1 / 8 = 1/8

4) P(B | A): The probability of B given A.
P(B | A) = P(A and B) / P(A)
P(A and B) = 1/8 (as calculated in the previous step)
P(A) = 1/2 (as calculated in step 1)
P(B | A) = (1/8) / (1/2) = 1/4 / 1/2 = 1/4 * 2/1 = 1/2

5) P(A | B): The probability of A given B.
P(A | B) = P(A and B) / P(B)
P(A and B) = 1/8 (as calculated in step 3)
P(B) = 1/4 (as calculated in step 2)
P(A | B) = (1/8) / (1/4) = 1/8 / 1/4 = 1/8 * 4/1 = 1/2

A and B are independent events if P(B | A) = P(B) or P(A | B) = P(A).
In this case, P(B | A) = 1/2 and P(B) = 1/4, so they are not equal.
Similarly, P(A | B) = 1/2 and P(A) = 1/2, so they are equal.
Since P(B | A) is not equal to P(B), events A and B are not independent.