Chapter 13.6, Problem 15E

To determine

To find: the probability of getting at most $3$ correct times from a bag if its lipstick is used $4$ time in a day.

Answer to Problem 15E

The probability of getting at most $3$ times correct is $\frac{65}{81}$ .

Explanation of Solution

Given:

The probability of getting the right color is $\frac{2}{3}$ .

Concept used:

Binomial probability refers to the probability of exactly $x$ successes on $n$ repeated trials in an experiment which has two possible outcomes common called the binomial experiment.

If the probability of success on an individual trial is $p$ and $q$ where $p+q=1$ or $q=1-p$ .

Then the binomial probability is:

  $C{}_x{p}^x{q}^{n-x}\text{ or }C{}_x{p}^x{\left(1-p\right)}^{n-x}$ .

The probability is minimum $0$ and maximum $1$ .

  $0\le p\left(A\right)\le 1.$

If $p\left(A\right)+p\left(B\right)=1$ then $p\left(B\right)=1-p\left(A\right)$ .

At least $x$ in the probability means considering the $x$ and greater than $x$ .

At most $x$ in the probability means considering the $x$ and lower than $x$ .

Calculation:

The probability of getting the right color is $\frac{2}{3}$ ,then the probability of getting the wrong color is $\frac{1}{3}$ .

  $\begin{array}{l}P\left(\text{ right color}\right)=\frac{2}{3}.\\ P\left(\text{not right color}\right)=\frac{1}{3}.\end{array}$

So, according to the given not more than $3$ means at most $3$ .

  $\begin{array}{l}P\left(\text{no more than }3\text{ times correct}\right)=1-P\left(\text{correct }4\text{ times}\right).\\ \Rightarrow 1-C{}_4{\left(\frac{2}{3}\right)}^4\times {\left(\frac{1}{3}\right)}^0.\\ \Rightarrow 1-1\times \frac{16}{81}\times 1.\\ \Rightarrow \frac{65}{81}.\end{array}$

Hence, the probability of getting at most $3$ times correct is $\frac{65}{81}$ .