Consider the following.
P(A) = 0.31 and P(B) = 0.41, and P(A and B) = 0.1271
(a) What is P(A | B)? (Give your answer correct to two decimal places.)
(b) What is P(B | A)? (Give your answer correct to two decimal places.)
(c) Are A and B independent?
yes no cannot be determined
(a)
P(A|B) conditional probability of A given B
=P(A∩B)/P(B) [∩ written as & c a p ; omit the spaces]
(b)
P(B|A)
=P(B∩A)/P(A)
(c) A and B are independent if and only if P(A∩B)=P(A)*P(B)
To calculate the values, we will use the formula for conditional probability:
(a) P(A | B) = P(A and B) / P(B)
(b) P(B | A) = P(A and B) / P(A)
Let's calculate each part step-by-step:
(a) P(A | B) = P(A and B) / P(B)
= 0.1271 / 0.41
≈ 0.31
Therefore, P(A | B) ≈ 0.31.
(b) P(B | A) = P(A and B) / P(A)
= 0.1271 / 0.31
≈ 0.41
Therefore, P(B | A) ≈ 0.41.
(c) To determine if A and B are independent, we compare their probabilities:
If P(A) * P(B) = P(A and B), then A and B are independent.
P(A) = 0.31
P(B) = 0.41
P(A and B) = 0.1271
Here, P(A) * P(B) = 0.31 * 0.41 ≈ 0.1271
Since P(A and B) is approximately equal to P(A) * P(B), we can conclude that A and B are independent.
Therefore, A and B are independent.
To find the answers to these questions, we will use the conditional probability formula:
P(A | B) = P(A and B) / P(B)
P(B | A) = P(A and B) / P(A)
Let's solve each part of the question:
(a) P(A | B):
We are given that P(A) = 0.31, P(B) = 0.41, and P(A and B) = 0.1271. Applying the formula, we get:
P(A | B) = P(A and B) / P(B)
P(A | B) = 0.1271 / 0.41
P(A | B) ≈ 0.31
So, P(A | B) ≈ 0.31. Therefore, the probability of A given B is approximately 0.31.
(b) P(B | A):
Using the same formula, we have:
P(B | A) = P(A and B) / P(A)
P(B | A) = 0.1271 / 0.31
P(B | A) ≈ 0.41
So, P(B | A) ≈ 0.41. Therefore, the probability of B given A is approximately 0.41.
(c) To determine whether A and B are independent, we need to check if P(A | B) = P(A) and P(B | A) = P(B). If these equations hold true, then A and B are independent.
In this case, we have:
P(A | B) ≈ 0.31 and P(A) = 0.31
P(B | A) ≈ 0.41 and P(B) = 0.41
Since P(A | B) = P(A) and P(B | A) = P(B), it means that A and B are independent. Therefore, the answer is: yes, A and B are independent.