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
$$ \begin{aligned} P(4\leq X\leq 8) & =P(X=4)+P(X=5)\\ &\quad +P(X=6)+P(X=7)+P(X=8)\\ &= 0.1886+0.2276\\ &\quad +0.2003+0.1295+0.0611\\ &=0.8071 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
