Answer :
To find the probability that there are two couples that fit the witness's description given that there is at least one, you can use the following formula:
P(A | B) = P(A and B) / P(B)
where A is the event "there are two couples that fit the witness's description," and B is the event "there is at least one couple that fits the witness's description."
To calculate P(A and B), you can use the formula:
P(A and B) = P(A) * P(B | A)
where P(B | A) is the probability that there is at least one couple that fits the witness's description given that there are two such couples.
In this case, P(A) is the probability that there are two couples that fit the witness's description out of the 5,000,000 couples in the Los Angeles area. There are a total of (5,000,000 choose 2) = 25,000,000,000 pairs of couples, so the probability that a randomly chosen pair fits the witness's description is 1/12,000,000. Therefore, P(A) is (1/12,000,000)^2 = 1/144,000,000,000,000.
Since there are two couples that fit the witness's description, the probability that there is at least one such couple is 1. Therefore, P(B | A) = 1.
Plugging these values into the formula for P(A and B), we get:
P(A and B) = (1/144,000,000,000,000) * 1 = 1/144,000,000,000,000
The probability that there is at least one couple that fits the witness's description is 1, since we are given that there is at least one such couple. Therefore, P(B) = 1.
Plugging these values into the formula for P(A | B), we get:
P(A | B) = (1/144,000,000,000,000) / 1 = 1/144,000,000,000,000
This is the probability that there are two couples that fit the witness's description given that there is at least one such couple
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