69% of men consider themselves professional baseball fans. You randomly select 10 men and ask each if he considers himself a professional baseball fan. Find the probability that the number who consider themselves baseball fans is
(a) exactly five,
(b) greater than or equal to six, and
(c) less than four.
Solution
Here $X$ denote the number of men who consider themselves professional baseball fans.
$p$ be the probability that men who consider themselves professional baseball fans.
Given that $p=0.69$ and $n =10$. The random variable $X\sim B(10, 0.69)$.
The probability mass function of $X$ is
$$
P(X=x) = \binom{10}{x} (0.69)^x (1-0.69)^{10-x}, \; x=0,1,\cdots, 10.
$$
(a) The probability that the number who consider themselves baseball fans is exactly five is
$$ \begin{aligned} P(X= 5) & =\sum_{x=0}^{5} P(x)-\sum_{x=0}^{4} P(x)\\\\ & = 0.1128\\ \end{aligned} $$
(b) The probability that the number who consider themselves baseball fans is greater than or equal to six is
$$ \begin{aligned} P(X\geq 6) & =\sum_{x=6}^{10} P(x)\\ & =\sum_{x=6}^{10}\binom{10}{x}(0.69)^x(1-0.69)^{10-x}\\ & = \binom{10}{6} (0.69)^{6} (1-0.69)^{10-6}+\binom{10}{7} (0.69)^{7} (1-0.69)^{10-7}\\ & \quad +\binom{10}{8} (0.69)^{8} (1-0.69)^{10-8}+\binom{10}{9} (0.69)^{9} (1-0.69)^{10-9}\\ &\quad +\binom{10}{10} (0.69)^{10} (1-0.69)^{10-10}\\ & = 0.2093 +0.2662+0.2222+0.1099+0.0245\\ & = 0.8321 \end{aligned} $$
(c) The probability that the number who consider themselves baseball fans is less than four is
$$ \begin{aligned} P(X< 4) & =\sum_{x=0}^{3} P(x)\\ & =\sum_{x=0}^{3}\binom{10}{x}(0.69)^x(1-0.69)^{10-x}\\ & = \binom{10}{0} (0.69)^{0} (1-0.69)^{10-0}+\binom{10}{1} (0.69)^{1} (1-0.69)^{10-1}\\ & \quad +\binom{10}{2} (0.69)^{2} (1-0.69)^{10-2}+\binom{10}{3} (0.69)^{3} (1-0.69)^{10-3}\\ & = 0 +0.0002+0.0018+0.0108\\ & = 0.0129 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
