According to a recent survey, outside of their own family members, 26% of adult Americans have no close friend to confide in. If this is the prevailing probability today, find the probability that in a random sample of n = 5 adults

(a) two or more have no close friend.

(b) at most two have no close friend.

(c) Find the expected number of persons who have no close friend.

#### Solution

Here $X$ denote the number of adult Americans who have no close friend to confide.

$p$ be the probability that adult American who have no close friend to confide.

Given that $p=0.26$ and $n =5$. Thus$X\sim B(5, 0.26)$.

The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{5}{x} (0.26)^x (1-0.26)^{5-x},\\ &\quad x=0,1,\cdots, 5. \end{aligned} $$ `

(a) The probability that 2 or more have no close friend is

` $$ \begin{aligned} P(X\geq 2) & =1-P(X\leq 1)\\ &= 1-\sum_{x=0}^{1} P(x)\\ &=1-\big(P(X=0)+P(X=1)\big)\\ &= 1- (0.2219+0.3898)\\ & = 1-0.6117 \\ & = 0.3883 \\ \end{aligned} $$ `

(b) The probability that at most two have no close friend is

` $$ \begin{aligned} P(X\leq 2) & =\sum_{x=0}^{2} P(x)\\ & =\sum_{x=0}^{2}\binom{5}{x}(0.26)^x(1-0.26)^{5-x}\\ &= P(X=0)+P(X=1) + P(X=1)\\ & = 0.2219+0.3898+0.2739\\ &=0.8857. \end{aligned} $$ `

(c) The expected number of persons who have no close friend is `$E(X) = n*p = 5 * 0.26 = 1.3$`

.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators