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 0.157.

Consider your next 30 purchases that produce a game piece. Calculate the following:

Round your answers to at least three decimal places.

a) What is the probability that you win 4 prizes? 

b) What is the probability that you win more than 6 prizes? 

c) What is the probability that you win between 3 and 6 (inclusive) prizes? 

d) What is the probability that you win 2 prizes or fewer? 


This problem involves calculating probabilities using the **binomial distribution**, which applies when you have a fixed number of independent trials, each with the same probability of success.

The binomial distribution formula is:

\[
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

where:
- \( n \) = number of trials (purchases) = 30,
- \( p \) = probability of success (winning a prize) = 0.157,
- \( k \) = number of successes (wins),
- \( \binom{n}{k} \) = binomial coefficient (number of ways to choose \( k \) successes from \( n \) trials).

Let's solve each part of the question step by step.

---

### a) Probability that you win exactly 4 prizes:
We need to calculate \( P(X = 4) \).

\[
P(X = 4) = \binom{30}{4} (0.157)^4 (1 - 0.157)^{30 - 4}
\]

---

### b) Probability that you win more than 6 prizes:
We need to calculate \( P(X > 6) \), which is:

\[
P(X > 6) = 1 - P(X \leq 6)
\]

First, calculate \( P(X \leq 6) \), which is the sum of the probabilities for \( X = 0, 1, 2, 3, 4, 5, 6 \).

---

### c) Probability that you win between 3 and 6 prizes (inclusive):
We need to calculate \( P(3 \leq X \leq 6) \), which is:

\[
P(3 \leq X \leq 6) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)
\]

---

### d) Probability that you win 2 prizes or fewer:
We need to calculate \( P(X \leq 2) \), which is the sum of the probabilities for \( X = 0, 1, 2 \).

Now, I will calculate these probabilities.Here are the probabilities based on the calculations:

a) The probability of winning exactly 4 prizes is approximately **0.196**.

b) The probability of winning more than 6 prizes is approximately **0.181**.

c) The probability of winning between 3 and 6 prizes (inclusive) is approximately **0.690**.

d) The probability of winning 2 prizes or fewer is approximately **0.129**.

Let me know if you'd like any further details or have any questions.

Here are 5 follow-up questions you might find interesting:
1. How would the probabilities change if the probability of winning a prize was 0.2 instead of 0.157?
2. What is the expected number of prizes you would win out of 30 purchases?
3. What is the variance and standard deviation of the number of prizes won?
4. What is the probability of winning exactly 0 prizes?
5. What is the probability of winning at least 1 prize out of 30 purchases?

**Tip:** The binomial distribution is ideal when each trial has only two possible outcomes (win or lose).

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 0.157.

Consider your next 30 purchases that produce a game piece. Calculate the following:

a) What is the probability that you win 4 prizes?

b) What is the probability that you win more than 6 prizes?

c) What is the probability that you win between 3 and 6 (inclusive) prizes?

d) What is the probability that you win 2 prizes or fewer?

Calculate the probability of winning prizes in a fast-food promotional game using the binomial distribution. Solve for the probability of winning exactly 4 prizes, more than 6 prizes, between 3 to 6 prizes, and 2 or fewer prizes out of 30 purchases with a win probability of 0.157. Suitable for advanced high school or college-level students.

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