Chapter 4.5, Problem 14E

Football Team Selection A football team consists of 20 freshmen and 20 sophomores, 15 juniors, and 10 seniors. Four players are selected at random to serve as captains. Find the probability that

a. All 4 are seniors

b. There is 1 each: freshman, sophomore, junior, and senior

c. There are 2 sophomores and 2 freshmen

d. At least 1 of the students is a senior

To determine

To obtain: The probability that all 4 players are seniors.

Answer to Problem 14E

The probability that all 4 players are seniors is 0.0003.

Explanation of Solution

There are 20 freshmen, 20 sophomores, 15 juniors and 10 seniors. The players selected at random to serve as captains is 4.

Calculation:

Combinations:

A combination is an arrangement of n objects in r ways where the order of arrangement is not considered.

${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}$

Let event A denote that all 4 players are seniors.

The number of ways for selecting 4 players are seniors is $C{}_4$

The total number of ways for selecting players is. $C{}_4$

The probability that all 4 players are seniors is,

$\begin{array}{c}P\left(A\right)=\frac{{}_{10}{C}_4}{{}_{65}{C}_4}\\ =\frac{210}{677,040}\\ =0.0003\end{array}$

Thus, the probability that all 4 players are seniors is 0.0003.

To determine

To obtain: The probability that there is 1 player from freshman, sophomore, junior and senior.

Answer to Problem 14E

The probability that there is 1 player from freshman, sophomore, junior and senior is 0.089.

Explanation of Solution

Calculation:

Let event B denote that there is 1 player from freshman, sophomore, junior and senior.

The number of ways for selecting 1 player from freshman, sophomore, junior and senior is $C{}_1\times C{}_1\times C{}_1\times C{}_1$.

The total number of players is 65.

The total number of ways for selecting players is $C{}_4$.

The probability that there is 1 player from freshman, sophomore, junior and senior is,

$\begin{array}{c}P\left(B\right)=\frac{C{}_1\times C{}_1\times C{}_1\times C{}_1}{{}_{65}{C}_4}\\ =\frac{20\times 20\times 15\times 10}{677,040}\\ =\frac{60,000}{677,040}\\ =0.089\end{array}$

Thus, the probability that there is 1 player from freshman, sophomore, junior and senior is 0.089.

To determine

To obtain: The probability that there are 2 sophomores and 2 freshmen.

Answer to Problem 14E

The probability that there are 2 sophomores and 2 freshmen is 0.053.

Explanation of Solution

Calculation:

Let event C denote that there are 2 sophomores and 2 freshmen.

The number of ways for selecting 2 sophomores and 2 freshmen is $C{}_2\times C{}_2$

The total number of players is 65.

The total number of ways for selecting players is $C{}_4$.

The probability that there are 2 sophomores and 2 freshmen is,

$\begin{array}{c}P\left(C\right)=\frac{C{}_2\times C{}_2}{{}_{65}{C}_4}\\ =\frac{190\times 190}{677,040}\\ =\frac{36,100}{677,040}\\ =0.053\end{array}$

Thus, the probability that there are 2 sophomores and 2 freshmen is 0.053.

To determine

To obtain: The probability that at least 1 of the student is a senior.

Answer to Problem 14E

The probability that at least 1 of the student is a senior is 0.496.

Explanation of Solution

Calculation:

Let event A denote that at least 1 of the student is a senior.

The number of ways for none of the student is a senior is $C{}_4$

The total number of ways for selecting players is $C{}_4$.

The probability that at least 1 of the student is a senior is,

$\begin{array}{c}P\left(A\right)=1-\frac{{}_{55}{C}_4}{{}_{65}{C}_4}\\ =1-\frac{341,055}{677,040}\\ =1-0.504\\ =0.496\end{array}$

Thus, the probability that at least 1 of the student is a senior is 0.496.