A well-mixed cookie dough will produce cookies with a mean of 8 chocolate chips apiece. What is the probability of getting a cookie with at least 4 chips?

#### Solution

Let $X$ denote the number chocolate chips. The mean number of chocolate chips a piece is 8, i.e., $E(X)=\lambda = 8$.

The random variable $X\sim P(8)$.

The probability mass function of Poisson distribution with $\lambda =8$ is

` $$ \begin{aligned} P(X=x) &= \frac{e^{-8}(8)^x}{x!},\\ \quad x=0,1,2,\cdots \end{aligned} $$ `

The probability of getting a cookie with at least 4 chips is

` $$ \begin{aligned} P(X\geq 4) &=1-P(X\leq 3)\\ &= 1- \sum_{x=0}^{3}P(X=x)\\ &=1-\big(P(X=0)+P(X=1)+P(X=2)+P(X=3)\big)\\ &= 1-\bigg(\frac{e^{-8}8^{0}}{0!}+\frac{e^{-8}8^{1}}{1!}\\ &\quad +\frac{e^{-8}8^{2}}{2!}+\frac{e^{-8}8^{3}}{3!}\bigg)\\ &= 1-\bigg(0.0003+0.0027\\ &\quad +0.0107+0.0286\bigg)\\ &= 0.9576 \end{aligned} $$ `

#### Further Reading

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