Q.4.74 - A First Course in Probability

Question

An interviewer is given a list of people she can interview. If the interviewer needs to interview 5 people, and if each person (independently) agrees to be interviewed with probability 2 3 , what is the probability that her list of people will enable her to obtain her necessary number of interviews if the list consists of

 (a) 5 people and

 (b) 8 people? For part (b), what is the probability that the interviewer will speak to exactly

 (c) 6 people and 

(d) 7 people on the list? 

Step-by-Step Solution

Answer

In the given information the answer of part (a) is $0.132$ , part (b) is $0.7414$ , part (c) is $0.2731$ and part (d) is  $0.1561$

Step 1:Given Information (Part-a)

If the interviewer needs to interview 5 people,and if each person agrees to be interviewed with probability of 2/3.

consider X is the random variable that represents the number of people that agree to be interviewed .

Step2: Calculation (Part-a)

$P(X=5)=\left(\begin{array}{l}n\\ 5\end{array}\right)p^5(1-p{)}^{n-5}$

               $=\left(\begin{array}{l}5\\ 5\end{array}\right){\left(\frac{2}{3}\right)}^5{\left(1-\frac{2}{3}\right)}^{5-5}$

                 $=1\times {\left(\frac{2}{3}\right)}^5{\left(\frac{1}{3}\right)}^0$

                  $=0.132$

Step 3:Final Answer (Part-a)

The final answer is $P\left(X=5\right)$ $0.132$

Step 4:Given Information (Part-b)

Consider Y is the random variable that represents the number of people that agree to be interviewed .

The number of people is 8.

The probability of success is 2/3.

$Y\ge 5$

Step 5 :Calculation (Part-b)

$P(Y\ge 5)=\sum _{1=5}^{3}P(Y=k)$

              $=\left[\begin{array}{l}P(Y=5)+P(Y=6)+\\ P(Y=7)+P(Y=8)\end{array}\right]$

                  $=$ $0.7414$

              

              

                    

Step 6:Final Answer (Part-b)

The final answer is $P\left(\left(Y\ge 5\right)\right)=$  $0.7414$

Step 7:Given Information(Part-c)

Consider Y is the random variable that represents the number of people . the list consists of 6 people. the probability of success is 2/3

Step 8: Calculation (Part-c)

$P(Y=6)=\left(\begin{array}{l}n\\ 6\end{array}\right)p^6(1-p{)}^{n-6}$

                $=\left(\begin{array}{l}8\\ 6\end{array}\right){\left(\frac{2}{3}\right)}^6{\left(1-\frac{2}{3}\right)}^{8-6}$

                $=0.2731$

Step 9:Final Answer (Part-c)

The final answer is $P\left(Y=6\right)=$$0.2731$

Step 10:Given Information (Part-d)

Consider Y is the random variable that represents the number of people . the list consists of 7 people. the probability of success is 2/3

Step 11 : Calculation (Part-d)

                $=\left(\begin{array}{l}8\\ 7\end{array}\right){\left(\frac{2}{3}\right)}^7{\left(1-\frac{2}{3}\right)}^{8-7}$

                  $=0.1561$

Step 12: Final Answer(Part-d)

The final answer is $0.1561$