Chapter 11.4, Problem 27PPS

a.

To determine

What is the probability that the same person has to wash the dishes $3$ times in a given week?

Answer to Problem 27PPS

  $\boxed{0.0816or8.16\%}$

Explanation of Solution

Given information:

Six roommates randomly select someone to wash the dishes each day.

What is the probability that the same person has to wash the dishes $3$ times in a given week?

Calculation:

The probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P(k){=}^n{C}_k{(p)}^k{(q)}^{n-k}$

Here $p$ represents the probability of success and $q$ represents the probability of failure.

Hence there are $6$ roommates, the probability of a roomamte getting selected is,

  $\begin{array}{l}p=\frac{1}{6}\\ =0.17\end{array}$

The probability of failure is,

  $\begin{array}{l}q=1-p\\ =1-0.17\\ =0.83\end{array}$

Substitute $p=0.17,q=0.83,k=3,n=7$ in the above formula to find the probability that the same person has to wash dishes $3$ times in a week, that is $7$ days,

  $\begin{array}{l}P(3){=}^7{C}_3{(}^{0.17}{(}^{0.83}\\ =0.0816\\ =8.16\%\end{array}$

Hence, the probability that the same person has to wash dishes $3$ times in a week, that is $7$ days.

  $\boxed{0.0816or8.16\%}$

b.

To determine

What method can the roommates use to select who washes the dishes each day?

Answer to Problem 27PPS

The person whose numbers turns up will wash the dishes that day.

Explanation of Solution

Given information:

Six roommates randomly select someone to wash the dishes each day.

What method can the roommates use to select who washes the dishes each day?

Calculation:

To make a fair decision, the random selection can be done by assigning each number from $1$ through $6$ to each person and then rolling a dice.

Hence, the person whose numbers turns up will wash the dishes that day.