Question

A large fast-food restaurant is having a promotional game where game pieces can be found on various products. Customers can win food or cash prizes. According to the company, the probability of winning a prize (large or small) with any eligible purchase is 0.199. Consider your next 13 purchases that produce a game piece. Calculate the following: Express your answer as a decimal with 4 decimal places. a) What is the probability that you win 3 prizes?

Answer 1

To calculate this, we will use the concept of probability using the binomial distribution.

A random experiment follows a binomial distribution if:

1) The experiment consists of n independent trials.
2) Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
3) The probability of success, denoted by P, is the same on every trial.

Given:
1) The probability of winning a prize (success probability or p) = 0.199.
2) Number of purchases/trials (n) = 13.
3) The number of prizes won (number of successes or k) = 3.

The probability, P(X=k), that an experiment of n trials results in exactly k successes is given by the formula:

P(X = k) = (n choose k) * (p^k) * (1 - p)^(n - k)

Where:
1) (n choose k) represents the number of combinations of n items taken k at a time.
2) p^k is the probability of getting k successes.
3) (1 - p)^(n - k) is the probability of getting n - k failures.

So, we substitute given values into this formula and calculate the probability.

After calculations, the estimated value will be 0.2450, which means there is 24.5% chance that you will win 3 prizes on your 13 eligible purchases.