Question

The town of Pleasantville is going to form a public safety committee. Pleasantville already has a seven-person town council, a five-person citizen advisory board; the police force is made up of ten officers. There is no overlap between the members of the town council, the citizen advisory board, and the police force.
Express all probabilities as decimals, rounded to six places.

14) If the committee must include three members from the town council, two members from the citizen advisory board, and three members from the police force, how many different ways can the committee be formed?

b) Suppose the public safety committee is made up as described in #14, and members are to be chosen at random from each of the town council, advisory board, and police force. If Tayler is on the town council, what is the probability they will be selected for the committee?

c) Suppose the public safety committee is made up as described in #14, and members are to be chosen at random from each of the town council, advisory board, and police force. If Casey is on the citizen advisory board, what is the probability they will be selected for the committee?

d) Suppose the public safety committee is made up as described in #14, and members are to be chosen at random from each of the town council, advisory board, and police force. If Pat is a police officer, what is the probability they will be selected for the committee?

Answer 1

P(Event) = Number of favorable outcomes/Total Number of outcomes

Please note ⁿC_x = n! / [(n-x)!*x!]

Town Council = 7, Citizen Advisory = 5 and Police Force = 10. Total = 7 + 5 + 10 = 22

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(a) 3 from town council = 7C3 = 35, 2 from advisory board = 5C2 = 10 and 3 from Police force = 10C3 = 120

Therefore total ways = 35 * 10 * 120 = 42,000

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(b) Total outcomes = 42000

If Taylor from the town council is selected, it means we need 2 more from the remaining 6 members of the town council, and other requirement being the same (2 from citizen and 3 from police)

2 from town council = 6C2 = 15, 2 from advisory board = 5C2 = 10 and 3 from Police force = 10C3 = 120

Therefore total ways = 15 * 10 * 120 = 18000

Therefore the required probability = 18000 / 42000 = 3 / 7 = 0.4286

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(c) Total outcomes = 42000

If Casey from the citizen advisory is selected, it means we need 1 more from the remaining 4 members of the citizen advisory, and other requirement being the same (3 from town and 3 from police)

3 from town council = 7C3 = 35, 1 from advisory board = 4C1 = 4 and 3 from Police force = 10C3 = 120

Therefore total ways = 35 * 4 * 120 = 16800

Therefore the required probability = 16800 / 42000 = 2 / 5 = 0.4

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(d) Total outcomes = 42000

If Pat from the Police force is selected, it means we need 2 more from the remaining 9 members of the police force, and other requirement being the same (3 from town and 2 from citizen advisory)

3 from town council = 7C3 = 35, 2 from advisory board = 6C2 = 15 and 2 from Police force = 9C2 = 36

Therefore total ways = 35 * 15 * 36 = 18900

Therefore the required probability = 18900 / 42000 = 9 / 20 = 0.45

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