Question

The probability that a bakery customer will order a birthday cake is .04.
a) What is the expected number of customers until the first birthday cake is ordered?
b) What is the probability the first cake order occurs within the first 20 customers? (Round your answer to 4 decimal places.)

Answer 1

Answer on Question #56184 – Math – Statistics and Probability

The probability that a bakery customer will order a birthday cake is .04.

a) What is the expected number of customers until the first birthday cake is ordered?

b) What is the probability the first cake order occurs within the first 20 customers? (Round your answer to 4 decimal places.)

Solution

The geometric distribution describes the number of Bernoulli trials until the first success is observed.

$$P(X \leq x) = 1 - (1 - \pi)^x$$

Let $X$ is the number of customers until the first birthday cake is ordered. Then, $X$ has a geometric distribution with $\pi = 0.04$.

a) The expected number of customers until the first birthday cake is ordered is

$$\mu = \frac{1}{\pi} = \frac{1}{0.04} = 25 \text{ customers}.$$

b) The probability the first cake order occurs within the first 20 customers is

$$P(X \leq 20) = 1 - (1 - 0.04)^{20} = 0.5580.$$

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