Zerrilloh6

Answered

2021-12-19

An insurance salesman sells policies to 5 men the prob. that a mean will be a live in 30 years is $\frac{2}{3}$ . Find the prob. that in 30 years: a) all 5 men , b) at least 3 men , c) only 2 men

Answer & Explanation

ol3i4c5s4hr

Expert

2021-12-20Added 48 answers

Let X be the number of men will be alive and n be sample number of men.

From the given information, probability that a men will be alive in 30 years is$\frac{2}{3}\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}n=5$ .

Here, men are independent and probability of success is constant. Hence, X follows binomial distribution with parameters$n=5\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}p=\frac{2}{3}$ .

The probability mass function of binomial random variable X is

$$P(X=x)=(\begin{array}{c}n\\ x\end{array}){p}^{x}(1-p{)}^{n-x};x=0,1,...,n$$

Step 3

a. The probability that in 30 years all 5 men will be alive is

$$P(X=5)=(\begin{array}{c}5\\ 5\end{array})(\frac{2}{3}{)}^{5}(1-\frac{2}{3}{)}^{5-5}$$

$=0.1317$

Thus, the probability that in 30 years all 5 men will be alive is 0.1317.

Step 4

b. The probability that in 30 years at least 3 men will be alive is

$P(X\ge 3)=P(X=3)+P(X=4)+P(X=5)$

$$=(\begin{array}{c}5\\ 3\end{array})(\frac{2}{3}{)}^{3}(1-\frac{2}{3}{)}^{5-3}+(\begin{array}{c}5\\ 4\end{array})(\frac{2}{3}{)}^{4}(1-\frac{1}{2}{)}^{5-4}+(\begin{array}{c}5\\ 5\end{array})(\frac{2}{3}{)}^{5}(1-\frac{2}{3}{)}^{5-5}$$

$=0.3292+0.3292+0.1317$

$=0.7901$

The probability that in 30 years at least 3 men will be alive is 0.7901.

Step 5

c. The probability that in 30 years all only 2 men will be alive is

$$P(X=2)=(\begin{array}{c}5\\ 2\end{array})(\frac{2}{3}{)}^{2}(1-\frac{2}{3}{)}^{5-2}$$

$=0.1646$

Thus, the probability that in 30 years all only 2 men will be alive is 0.1646.

From the given information, probability that a men will be alive in 30 years is

Here, men are independent and probability of success is constant. Hence, X follows binomial distribution with parameters

The probability mass function of binomial random variable X is

Step 3

a. The probability that in 30 years all 5 men will be alive is

Thus, the probability that in 30 years all 5 men will be alive is 0.1317.

Step 4

b. The probability that in 30 years at least 3 men will be alive is

The probability that in 30 years at least 3 men will be alive is 0.7901.

Step 5

c. The probability that in 30 years all only 2 men will be alive is

Thus, the probability that in 30 years all only 2 men will be alive is 0.1646.

lovagwb

Expert

2021-12-21Added 50 answers

1) Probability that all men will be alive: $P={\left(\frac{2}{3}\right)}^{5}=\frac{32}{243}$ .

2) Probability that at least 3 men will be alive:$P(x\ge 3)=C(5,3)\cdot {\left(\frac{2}{3}\right)}^{3}\cdot {\left(\frac{1}{3}\right)}^{2}+C(5,4)\cdot {\left(\frac{2}{3}\right)}^{4}\cdot {\left(\frac{1}{3}\right)}^{1}+C(5,5)\cdot {\left(\frac{2}{3}\right)}^{5}=\frac{80}{243}+\frac{80}{243}+\frac{32}{243}=\frac{192}{243}$ .

3) Only two men will be alive:$P(x=2)=C(5,2)\cdot {\left(\frac{2}{3}\right)}^{2}\cdot {\left(\frac{1}{3}\right)}^{3}=\frac{40}{243}$ .

4) At least 1 man will be alive:$P(x\ge 1)=1-P(x=0)=1-C(5,0)\cdot {\left(\frac{1}{3}\right)}^{5}=\frac{242}{243}$ .

2) Probability that at least 3 men will be alive:

3) Only two men will be alive:

4) At least 1 man will be alive:

nick1337

Expert

2021-12-28Added 573 answers

Step-by-step explanation:

a)

b)

c)

d)

Most Popular Questions