Question

A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows:

A: The number is even
B: The number is less than 7

Find P(A | B) and P(B | A).

Answer 1

To find P(A|B), we calculate the probability of event A (the number is even) given that event B (the number is less than 7) has already occurred. The probability of A given B is 1/2, and the probability of B given A is 3/5.

Step-by-step explanation:

To find P(A|B), we need to find the probability of event A (the number is even) given that event B (the number is less than 7) has already occurred. To do this, we first determine the probability of event B. Since there are 6 numbers less than 7 (1, 2, 3, 4, 5, 6) out of a total of 10 numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), the probability of event B is 6/10.

Next, we determine the probability of event A and B occurring together (A and B) by finding the numbers that satisfy both conditions. There are 3 even numbers less than 7 (2, 4, 6), so the probability of A and B is 3/10.

Finally, we use the formula P(A|B) = P(A and B) / P(B) to calculate P(A|B). Plugging in the values, we get P(A|B) = (3/10) / (6/10) = 1/2.

To find P(B|A), we can use the formula P(B|A) = P(A and B) / P(A). Plugging in the values, we get P(B|A) = (3/10) / (5/10) = 3/5.