# Solved (Free): Forty-three percent of adults would pay more for environmentally friendly products. You randomly select 12 adults

#### ByDr. Raju Chaudhari

Mar 29, 2021

Forty-three percent of adults would pay more for environmentally friendly products. You randomly select 12 adults. Find the probability that the number of adults who would pay more for environmentally friendly products is

a) exactly 4,
b) more than 4, and
c) between 4 and 8, inclusive.

#### Solution

Here $X$ denote the number of adults who pay more for environmentally friendly products.

$p$ be the percent of adults who pay more for environmentally friendly products.

Given that $p=0.43$ and $n =12$. Thus $X\sim B(12, 0.43)$.

The probability mass function of $X$ is
 \begin{aligned} P(X=x) &= \binom{12}{x} (0.43)^x (1-0.43)^{12-x},\\ &\quad x=0,1,\cdots, 12. \end{aligned}

$x$ $P(X=x)$
0 0.0012
1 0.0106
2 0.0442
3 0.1111
4 0.1886
5 0.2276
6 0.2003
7 0.1295
8 0.0611
9 0.0205
10 0.0046
11 0.0006
12 0.0000

(a) The probability that the number of adults who would pay more for environmentally friendly products is exactly 4 is

 \begin{aligned} P(X= 4) & =\binom{12}{4} (0.43)^{4} (1-0.43)^{12-4}\\\\ & = 0.1886\\ \end{aligned}

(b) The probability that the number of adults who would pay more for environmentally friendly products is more than 4 is

 \begin{aligned} P(X\geq 4) & =1-P(X\leq 3)\\ &= 1-\sum_{x=0}^{3} P(x)\\ & = 1-\bigg(P(X=0)+P(X=1)\\ &\quad +P(X=2)+P(X=3)\bigg)\\ &= 1- \bigg(0.0012+0.0106\\ &\quad +0.0442+0.1111\bigg) \\ &= 1- 0.1671\\ & = 0.8329 \end{aligned}

(c) The probability that the number of adults who would pay more for environmentally friendly products is between 4 and 8 (inclusive) is

 \begin{aligned} P(4\leq X\leq 8) & =P(X=4)+P(X=5)\\ &\quad +P(X=6)+P(X=7)+P(X=8)\\ &= 0.1886+0.2276\\ &\quad +0.2003+0.1295+0.0611\\ &=0.8071 \end{aligned}