Chapter 13.2, Problem 8PPS

(a)

To determine

To find: Theprobability that Cecilia, Annie and Kimi are first $3$ gymnasts to perform in any order.

Answer to Problem 8PPS

The probability is $\frac{1}{35}$ .

Explanation of Solution

Given:

The table shows the finalist for a floor exercises competition.

  

Concept used:

The number of ways of arranging $n$ item in order $=n!$ .

The probability formula is used to compute the probability of an event to occur as:

  $P\left(A\right)=\frac{\text{Number of favorable outcome}}{\text{Total Number of favorable outcomes}}$ .

Calculation:

The number of possible outcomes in the number of arrangements of $7$ performers taken $7$ at time.

So, the number of possible outcomes is $7!=5040$ .

The first $3$ finalists can be arranged in $3!=6$ ways.

The rest of the finalists can be arranged $4!$ ways.

Therefore, the number of favourable outcomes is $4!\times 3!=24\times 6=144.$

The probability is $\frac{144}{5040}=\frac{1}{35}$ .

Hence, the probability is $\frac{1}{35}$ .

(b)

To determine

To find:the probability that Cecilia is first, Annie is second and Kimi is third.

Answer to Problem 8PPS

The probability is $\frac{1}{210}$ .

Explanation of Solution

Given:

  

Concept used:

The number of ways of arranging $n$ item in order $=n!$ .

The probability formula is used to compute the probability of an event to occur as:

  $P\left(A\right)=\frac{\text{Number of favorable outcome}}{\text{Total Number of favorable outcomes}}$ .

Calculation:

The number of possible outcomes is $5040$ .the number of favourable outcomes with Cecilia in first place, Annie second, and Kimi in third place is $1$ .after the first three places are set, the rest of the finalists can be arranged $4!$ ways.

Therefore, the probability is $\frac{24}{5040}=\frac{1}{210}$ .

Hence, the probability is $\frac{1}{210}$ .