There are 8 people taking part in a raffle.Ann, Bob, Elsa, Hans, Kira, Omar, Ravi, and Soo.Suppose that prize winners are randomly selected from the 8 people.Compute the probability of each of the following events.Event A: The first four prize winners are Soo, Bob, Kira, and Hans, regardless of order.Event B: Kira is the first prize winner, Bob is second, Omar is third, and Ann is fourth.Write your answers as fractions in simplest form.PA = PB =
Question
There are 8 people taking part in a raffle.Ann, Bob, Elsa, Hans, Kira, Omar, Ravi, and Soo.Suppose that prize winners are randomly selected from the 8 people.Compute the probability of each of the following events.Event A: The first four prize winners are Soo, Bob, Kira, and Hans, regardless of order.Event B: Kira is the first prize winner, Bob is second, Omar is third, and Ann is fourth.Write your answers as fractions in simplest form.PA = PB =
Solution
Event A: The first four prize winners are Soo, Bob, Kira, and Hans, regardless of order.
The total number of ways to select 4 winners out of 8 people is given by the combination formula C(n, r) = n! / [(n-r)!r!], where n is the total number of people, r is the number of winners, and '!' denotes factorial.
So, the total number of ways to select 4 winners out of 8 people is C(8, 4) = 8! / [(8-4)!4!] = 70.
Since the order of the winners does not matter in Event A, there is only 1 way to select Soo, Bob, Kira, and Hans as the winners.
Therefore, the probability of Event A, P(A), is the number of favorable outcomes divided by the total number of outcomes, which is 1/70.
Event B: Kira is the first prize winner, Bob is second, Omar is third, and Ann is fourth.
In this case, the order of the winners does matter. The total number of ways to select 4 winners out of 8 people in a specific order is given by the permutation formula P(n, r) = n! / (n-r)!, where n is the total number of people, r is the number of winners, and '!' denotes factorial.
So, the total number of ways to select 4 winners out of 8 people in a specific order is P(8, 4) = 8! / (8-4)! = 1680.
Since Kira, Bob, Omar, and Ann must win in a specific order in Event B, there is only 1 way for this to happen.
Therefore, the probability of Event B, P(B), is the number of favorable outcomes divided by the total number of outcomes, which is 1/1680.
So, P(A) = 1/70 and P(B) = 1/1680.
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