Every twentieth customer at a coffee shop gets a $5 gift card. Design a simulation for estimating the probability that a customer would need to visit the coffee shop at least 5 times to win a gift card.

The probability that a customer would visit the coffee shop at least 5 times to win a gift card is [tex](\frac {11}{12})^{n} - (\frac n{12}) * (\frac {11}{12})^{n - 1} - .......... - ^nC_4 * (\frac 1{12})^4 * (\frac {11}{12})^{n - 4}[/tex]

How to design the simulation?

From the question, we understand that every 12th customer gets a $5 gift card.

This means that the probability that a customer gets a gift card is:

p = 1/12

The probability that a customer would visit the coffee shop at least 5 times to win a gift card is calculated using the following complement rule:

P(x ≥ 5) = 1 - P(x <5)

The equation becomes

P(x ≥ 5) = 1 - P(0) - P(1) - P(2) - P(3) - P(4)

Using the binomial formula

[tex]P(x) = ^nC_x * p^x * (1 - p)^{n - x}[/tex]

The equation becomes

[tex]P(x \ge 5) = ^nC_0 * (\frac 1{12})^0 * (1 - \frac 1{12})^{n - 0} - ^nC_1 * (\frac 1{12})^1 * (1 - \frac 1{12})^{n - 1} - .......... - ^nC_4 * (\frac 1{12})^4 * (1 - \frac 1{12})^{n - 4}[/tex]

Simplify

[tex]P(x \ge 5) = (\frac {11}{12})^{n} - (\frac n{12}) * (\frac {11}{12})^{n - 1} - .......... - ^nC_4 * (\frac 1{12})^4 * (\frac {11}{12})^{n - 4}[/tex]

Hence, the estimate that a customer would visit at least 5 times is [tex]P(x \ge 5) = (\frac {11}{12})^{n} - (\frac n{12}) * (\frac {11}{12})^{n - 1} - .......... - ^nC_4 * (\frac 1{12})^4 * (\frac {11}{12})^{n - 4}[/tex]