In a major league baseball game, the average is 0.9 broken bat per game.

(a) Find the probability of no broken bats in a game. (Round your answer to 4 decimal places.)

(b) Find the probability of at least 2 broken bats in a game. (Round your answer to 4 decimal places.)

#### Solution

Let $X$ denote the number of broken bat per game. The average number of broken bat per game is 0.9, i.e., `$E(X) = \lambda = 0.9$`

.

`$X\sim P(0.9)$`

.

The probability mass function of Poisson distribution with `$\lambda =0.9$`

is

` $$ \begin{aligned} P(X=x) &= \frac{e^{-0.9}(0.9)^x}{x!},\; x=0,1,2,\cdots \end{aligned} $$ `

a) The probability of no broken bats in a game is $P(X =0)$

` $$ \begin{aligned} P(X = 0) &= \frac{e^{-0.9}0.9^{0}}{0!}\\ &= 0.4066 \end{aligned} $$ `

b) The probability of at least 2 broken bats in a game is $P(X\geq 2)$.

` $$ \begin{aligned} P(X\geq2) &= 1- P(X\leq 1)\\ &= 1- \big(P(X=0)+ P(X=1)\big)\\ &= 1- \bigg(\frac{e^{-0.9}0.9^{0}}{0!}+\frac{e^{-0.9}0.9^{1}}{1!}\bigg)\\ &= 1-\big(0.4066+0.3659\big)\\ &= 0.2275 \end{aligned} $$ `

#### Further Reading

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