Questions: Six stand-up comics, A, B, C, D, E, and F, are to perform on a single evening at a comedy club. The order of performance is determined by random selection. Find the probability that: a. Comic F will perform fourth. b. Comic A will perform third and Comic D will perform fourth. c. The comedians will perform in the following order: F, A, D, C, B, E. d. Comic B or Comic E will perform fourth. a. (Type a fraction. Simplify your answer.) b. (Type a fraction. Simplify your answer.) c. (Type a fraction. Simplify your answer.) d. (Type a fraction. Simplify your answer.)

$Six stand-up comics, A, B, C, D, E, and F, are to perform on a single evening at a comedy club. The order of performance is determined by random selection. Find the probability that: a. Comic F will perform fourth. b. Comic A will perform third and Comic D will perform fourth. c. The comedians will perform in the following order: F, A, D, C, B, E. d. Comic B or Comic E will perform fourth. a. (Type a fraction. Simplify your answer.) b. (Type a fraction. Simplify your answer.) c. (Type a fraction. Simplify your answer.) d. (Type a fraction. Simplify your answer.)$

Transcript text: Six stand-up comics, A, B, C, D, E, and F, are to perform on a single evening at a comedy club. The order of performance is determined by random selection. Find the probability that: a. Comic F will perform fourth. b. Comic A will perform third and Comic D will perform fourth. c. The comedians will perform in the following order: F, A, D, C, B, E. d. Comic B or Comic E will perform fourth. a. $\square$ (Type a fraction. Simplify your answer.) b. $\square$ (Type a fraction. Simplify your answer.) c. $\square$ (Type a fraction. Simplify your answer.) d. $\square$ (Type a fraction. Simplify your answer.)

Solution

Solution Steps

To solve these probability questions, we need to consider the total number of possible arrangements of the six comics and then determine the number of favorable outcomes for each specific scenario.

a. To find the probability that Comic F will perform fourth, we need to count the number of arrangements where F is in the fourth position and divide it by the total number of arrangements.

b. To find the probability that Comic A will perform third and Comic D will perform fourth, we need to count the number of arrangements where A is in the third position and D is in the fourth position, and divide it by the total number of arrangements.

c. To find the probability that the comedians will perform in the exact order F, A, D, C, B, E, we need to check if this specific arrangement is one of the possible arrangements and divide it by the total number of arrangements.

d. To find the probability that Comic B or Comic E will perform fourth, we need to count the number of arrangements where either B or E is in the fourth position and divide it by the total number of arrangements.

Step 1: Total Arrangements

The total number of arrangements of the six comics $ A, B, C, D, E, F $ is given by $ 6! $: \[ 6! = 720 \]

Step 2: Probability that Comic F will perform fourth

To find the probability that Comic F performs fourth, we fix F in the fourth position and arrange the remaining 5 comics. The number of arrangements is $ 5! $: \[ 5! = 120 \] Thus, the probability $ P(A) $ is: \[ P(A) = \frac{5!}{6!} = \frac{120}{720} = \frac{1}{6} \approx 0.1667 \]

Step 3: Probability that Comic A will perform third and Comic D will perform fourth

For Comic A to perform third and Comic D to perform fourth, we fix A in the third position and D in the fourth position, arranging the remaining 4 comics. The number of arrangements is $ 4! $: \[ 4! = 24 \] Thus, the probability $ P(B) $ is: \[ P(B) = \frac{4!}{6!} = \frac{24}{720} = \frac{1}{30} \approx 0.03333 \]

Step 4: Probability that the comedians will perform in the exact order F, A, D, C, B, E

There is only one specific arrangement for the order $ F, A, D, C, B, E $: \[ \text{Specific order} = 1 \] Thus, the probability $ P(C) $ is: \[ P(C) = \frac{1}{6!} = \frac{1}{720} \approx 0.00139 \]

Step 5: Probability that Comic B or Comic E will perform fourth

To find the probability that either Comic B or Comic E performs fourth, we can fix either B or E in the fourth position and arrange the remaining 5 comics. The number of arrangements for each case is $ 5! $, and since there are 2 cases: \[ 2 \times 5! = 240 \] Thus, the probability $ P(D) $ is: \[ P(D) = \frac{2 \times 5!}{6!} = \frac{240}{720} = \frac{1}{3} \approx 0.3333 \]