An old bus breaks down an average of 3 times per month. Using the Poisson probability distribution formula, find the probability that during the next month this bus will have
i) exactly 2 breakdowns;
ii) at most one breakdown.
Solution
Let $X$ denote the number of break downs per months. The average number of break downs is 3, i.e., $E(X) = \lambda = 3$.
$X\sim P(3)$.
The probability mass function of Poisson distribution with $\lambda =3$ is
$$ \begin{aligned} P(X=x) &= \frac{e^{-3}(3)^x}{x!},\; x=0,1,2,\cdots \end{aligned} $$
i) The probability that during the next month this bus will have exactly 2 breakdowns is $P(X =2)$
$$ \begin{aligned} P(X = 2) &= \frac{e^{-3}3^{2}}{2!}\\ &= 0.224 \end{aligned} $$
ii) The probability that during the next month this bus will have at most one breakdown is $P(X\leq 1)$.
$$ \begin{aligned} P(X\leq1) &= P(X=0)+ P(X=1)\\ &= \frac{e^{-3}3^{0}}{0!}+\frac{e^{-3}3^{1}}{1!}\\ &= 0.0498+0.1494\\ &= 0.1992 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
