Questions: There are 9 acts in a talent show. A comedian, a dancer, a guitarist, a juggler, a magician, a pianist, a singer, a violinist, and a whistler. A talent show host randomly schedules the 9 acts. Compute the probability of each of the following events. Event A: The comedian is first, the pianist is second, and the juggler is third. Event B: The first three acts are the violinist, the magician, and the singer, in any order. Write your answers as fractions in simplest form. P(A)= P(B)=

$There are 9 acts in a talent show. A comedian, a dancer, a guitarist, a juggler, a magician, a pianist, a singer, a violinist, and a whistler. A talent show host randomly schedules the 9 acts. Compute the probability of each of the following events. Event A: The comedian is first, the pianist is second, and the juggler is third. Event B: The first three acts are the violinist, the magician, and the singer, in any order. Write your answers as fractions in simplest form. P(A)= P(B)=$

Transcript text: There are 9 acts in a talent show. A comedian, a dancer, a guitarist, a juggler, a magician, a pianist, a singer, a violinist, and a whistler. A talent show host randomly schedules the 9 acts. Compute the probability of each of the following events. Event A: The comedian is first, the pianist is second, and the juggler is third. Event B: The first three acts are the violinist, the magician, and the singer, in any order. Write your answers as fractions in simplest form. \[ \begin{array}{l} P(A)= \\ P(B)= \end{array} \]

Solution

Solution Steps

Step 1: Calculate Total Permutations

The total number of ways to arrange 9 acts is given by $ 9! $: \[ 9! = 362880 \]

Step 2: Calculate Probability of Event A

For Event A, where the comedian is first, the pianist is second, and the juggler is third, there is only 1 favorable outcome. Thus, the probability $ P(A) $ is: \[ P(A) = \frac{1}{362880} \]

Step 3: Calculate Probability of Event B

For Event B, where the first three acts are the violinist, the magician, and the singer in any order, there are $ 3! = 6 $ favorable outcomes. Therefore, the probability $ P(B) $ is: \[ P(B) = \frac{6}{362880} = \frac{1}{60480} \]

Final Answer

\[ P(A) = \frac{1}{362880}, \quad P(B) = \frac{1}{60480} \] Thus, the final answers are: \[ \boxed{P(A) = \frac{1}{362880}, \quad P(B) = \frac{1}{60480}} \]

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