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QuestionIn a talent show with 8 acts, find the probabilities of: A) singer first, comedian second, dancer third, pianist fourth; B) any order of pianist, comedian, dancer, juggler in first four. Provide answers as simplified fractions. P(A)= P(A)= P(B)= P(B)=

Studdy Solution

STEP 1

Assumptions1. There are8 acts in total. . The acts are scheduled randomly.
3. The order of the acts matters.
4. Event A is that the singer is first, the comedian is second, the dancer is third, and the pianist is fourth.
5. Event B is that the first four acts are the pianist, the comedian, the dancer, and the juggler, in any order.

STEP 2

We first need to understand that the total number of ways to schedule8 acts is given by8 factorial (8!). This is because there are8 choices for the first act,7 for the second,6 for the third, and so on, until there is only1 choice for the last act.
Total ways=8!Total\, ways =8!

STEP 3

For event A, the first four acts are fixed. The remaining acts can be arranged in! ways. So, the total number of ways event A can occur is!.
Ways for A=!Ways\, for\, A =!

STEP 4

The probability of an event is given by the ratio of the number of ways the event can occur to the total number of outcomes. So, the probability of event A is(A)=Ways for ATotal ways(A) = \frac{Ways\, for\, A}{Total\, ways}

STEP 5

Substitute the values we calculated into the formula to find(A).
(A)=4!8!(A) = \frac{4!}{8!}

STEP 6

implify the fraction to find(A).
(A)=14×5×6××8=120160(A) = \frac{1}{4 \times5 \times6 \times \times8} = \frac{1}{20160}

STEP 7

For event B, the first four acts can be any of the pianist, the comedian, the dancer, and the juggler, in any order. This can be arranged in4! ways. The remaining4 acts can also be arranged in4! ways. So, the total number of ways event B can occur is4! *4!.
Ways for B=4!×4!Ways\, for\, B =4! \times4!

STEP 8

Substitute the values we calculated into the formula to find(B).
(B)=Ways for BTotal ways(B) = \frac{Ways\, for\, B}{Total\, ways}

STEP 9

Substitute the values we calculated into the formula to find(B).
(B)=4!×4!8!(B) = \frac{4! \times4!}{8!}

STEP 10

implify the fraction to find(B).
(B)=4×3×2××4×3×2×8×7×6×5×4×3×2×=420(B) = \frac{4 \times3 \times2 \times \times4 \times3 \times2 \times}{8 \times7 \times6 \times5 \times4 \times3 \times2 \times} = \frac{}{420}So, the probability of event A is/20160 and the probability of event B is/420.

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