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.
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.
Answers
GPT 3.5
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.
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.