Answer:
the probability that the two committees have at least one member in common is approximately 0.3622.
Explanation:
To find the probability that two committees have at least one member in common, we can use the complement rule, which states that the probability of an event happening is 1 minus the probability of the event not happening.
Let A be the event that the two committees have no members in common. To calculate the probability of A, we can first find the number of ways to choose three people out of 20 without any overlap, and then use this to find the total number of ways to choose two committees of three people each without any overlap:
Number of ways to choose 3 people out of 20 without overlap:
C(20,3) = (201918)/(321) = 1140
Number of ways to choose 3 people out of 20 with overlap:
C(20,3) - C(2,1)C(17,2) = 1140 - 2(2*136) = 680
The first term is the total number of ways to choose 3 people out of 20, and the second term is the number of ways to choose 3 people out of 20 such that one particular person is included. We multiply by 2 because there are two committees.
The total number of ways to choose two committees of three people each without any overlap is:
C(20,3) * C(17,3) = (1140*680) = 775200
Therefore, the probability that the two committees have no members in common is:
P(A) = 775200 / (C(20,3)*C(17,3)) = 0.6378
So the probability that the two committees have at least one member in common is:
P(at least one member in common) = 1 - P(A) = 1 - 0.6378 = 0.3622
Therefore, the probability that the two committees have at least one member in common is approximately 0.3622.