Given the following information answer questions a-d.
P(A)= 0.42
P(B)=0.36
A and B are independent Round all answer to the nearest 5 decimal places as needed.
a) Find P(A U B)
b) Find P ( A|B)
C) Find P (B|A)
d) Find P(AnB)
To answer the given questions, we need to understand some basic probability concepts. Here's a quick explanation of the concepts involved:
1. Union (A U B): The union of two events A and B represents the occurrence of either event A, event B, or both. In this case, we need to find the probability of event A or event B happening.
2. Conditional Probability (A|B): Conditional probability represents the probability of event A occurring given that event B has already occurred. In this case, we need to find the probability of event A happening, assuming event B has already occurred.
3. Independent Events: When two events A and B are independent, the occurrence of one event does not impact the occurrence of the other event. In other words, the probability of event A happening is not influenced by whether event B occurs, and vice versa.
With these concepts in mind, let's answer the questions:
a) To find P(A U B), we can use the formula:
P(A U B) = P(A) + P(B) - P(A and B)
Since A and B are independent, P(A and B) = P(A) * P(B)
P(A U B) = P(A) + P(B) - P(A) * P(B)
Substituting the given values:
P(A U B) = 0.42 + 0.36 - (0.42 * 0.36)
Calculate the result to get the answer.
b) To find P(A|B), we use the formula:
P(A|B) = P(A and B) / P(B)
Since A and B are independent, P(A and B) = P(A) * P(B)
P(A|B) = (P(A) * P(B)) / P(B)
P(A|B) = P(A)
Substituting the given value for P(A), we get the answer.
c) To find P(B|A), we use the same formula as in part (b):
P(B|A) = P(A and B) / P(A)
Since A and B are independent, P(A and B) = P(A) * P(B)
P(B|A) = (P(A) * P(B)) / P(A)
P(B|A) = P(B)
Substituting the given value for P(B), we get the answer.
d) To find P(A and B), we use the formula:
P(A and B) = P(A) * P(B)
Substituting the given values, calculate the result to get the answer.
Remember to round all answers to the nearest 5 decimal places as stated in the question.