Questions: There are 8 people taking part in a raffle. Ann, Bob, Elsa, Hans, Jim, Kira, Omar, 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 Kira, Omar, Soo, and Hans, regardless of order. Event B: Soo is the first prize winner, Omar is second, Bob is third, and Hans is fourth. Write your answers as fractions in simplest form. P(A)= P(B)=

There are 8 people taking part in a raffle. Ann, Bob, Elsa, Hans, Jim, Kira, Omar, 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 Kira, Omar, Soo, and Hans, regardless of order. Event B: Soo is the first prize winner, Omar is second, Bob is third, and Hans is fourth. Write your answers as fractions in simplest form. P(A)= P(B)=
Transcript text: There are 8 people taking part in a raffle. Ann, Bob, Elsa, Hans, Jim, Kira, Omar, 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 Kira, Omar, Soo, and Hans, regardless of order. Event B: Soo is the first prize winner, Omar is second, Bob is third, and Hans is fourth. Write your answers as fractions in simplest form. \[ \begin{array}{l} P(A)= \\ P(B)= \end{array} \] $\square$
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Solution

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Solution Steps

To solve this problem, we need to calculate the probabilities of two specific events in a raffle with 8 participants.

Event A
  • We need to find the probability that the first four prize winners are Kira, Omar, Soo, and Hans, regardless of order.
  • Calculate the number of favorable outcomes: the number of ways to arrange Kira, Omar, Soo, and Hans among the first four winners.
  • Calculate the total number of possible outcomes: the number of ways to choose any 4 winners from 8 participants.
  • The probability of Event A is the ratio of favorable outcomes to total outcomes.
Event B
  • We need to find the probability that Soo is the first prize winner, Omar is second, Bob is third, and Hans is fourth.
  • There is only one favorable outcome for this specific order.
  • Calculate the total number of possible outcomes: the number of ways to arrange all 8 participants.
  • The probability of Event B is the ratio of the single favorable outcome to the total outcomes.
Step 1: Calculate Favorable Outcomes for Event A

For Event A, we need to determine the number of ways to arrange Kira, Omar, Soo, and Hans among the first four winners. This is given by the number of permutations of 4 people, which is \(4!\).

\[ 4! = 24 \]

Step 2: Calculate Total Outcomes for Event A

The total number of ways to choose any 4 winners from 8 participants and arrange them is given by the combination of 8 choose 4, multiplied by the permutations of those 4 winners.

\[ \binom{8}{4} \times 4! = 70 \times 24 = 1680 \]

Step 3: Calculate Probability of Event A

The probability of Event A is the ratio of favorable outcomes to total outcomes.

\[ P(A) = \frac{24}{1680} = \frac{1}{70} \]

Step 4: Calculate Favorable Outcomes for Event B

For Event B, there is only one specific order: Soo is first, Omar is second, Bob is third, and Hans is fourth. Thus, there is only 1 favorable outcome.

\[ \text{Favorable outcomes for Event B} = 1 \]

Step 5: Calculate Total Outcomes for Event B

The total number of ways to arrange all 8 participants is given by the permutations of 8 people, which is \(8!\).

\[ 8! = 40320 \]

Step 6: Calculate Probability of Event B

The probability of Event B is the ratio of the single favorable outcome to the total outcomes.

\[ P(B) = \frac{1}{40320} \]

Final Answer

\[ \begin{array}{l} P(A)= \frac{1}{70} \\ P(B)= \frac{1}{40320} \end{array} \] $\square$

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