Question

There are 9 acts in a talent show.

Answer 1

Answer:

Event A. P(A) = 1/28

Event B. P(B) = 1/56

Step-by-step explanation:

Event A. We have 3! ways of arranging the juggler, acrobat and pianist and ⁶P₁ ways of arranging the rest of the performers. We ave ⁹P₁ ways of arranging all the performers.

So, P(A) = 3!⁶P₁/⁹P₁ = 3!6!/9! =  3!6!/(9 × 8 × 7 × 6!) = 6/(3 × 8 × 7) = 1/28

Event B. We have ³C₁ ways of combining the violinist, acrobat and comedian in any order and ⁶P₁ ways of arranging the rest of the performers. We ave ⁹P₁ ways of arranging all the performers.

So, P(B) = ³C₁ × ⁶P₁/⁹P₁ = 3!6!/9! =  3 × 6!/(9 × 8 × 7 × 6!) = 3/(3 × 8 × 7) = 1/56

Answer 2

Answer:

Event A: The juggler is first, the acrobat is second, and the pianist is third = 1/84

Event B: The first three acts are the violinist, the acrobat, and the comedian, in any order = 1/168

Step-by-step explanation:

We have nine acts ready to perform in a Talent show.

Event A: The juggler is first, the acrobat is second, and the pianist is third.

Therefore, the probability of the Juggler performing first, the acrobat second and the pianist third is given as:

= 3! = 3× 2× 1

Followed by other acts = ⁶P₁/⁹P₁

Therefore, the probability of Event A happening is

3!× ⁶P₁/⁹P₁ = 3! × 6!/9! =  3×2× 1 (6×5×4×3×2× 1) /(9 × 8 × 7 × 6 × 5×4×3×2× 1) ) = 6/(9 × 8 ×7) = 1/84

Event B: The first three acts are the violinist, the acrobat, and the comedian, in any order

Arranging the first 3 acts in any order =

³C₁

Followed by other acts = ⁶P₁/⁹P₁

Hence, the probability of event B happening

= ³C₁ × ⁶P₁/⁹P₁ = 3 × 6!/9! =  3 × 6!/(9 × 8 × 7 × 6!) = 3/(9 × 8 × 7) = 1/168