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Let's break down and solve each part of the problem.

### Problem Recap:
A teacher provides pizza for his class if they earn an A-average on any test. The probability of the class getting an A-average on one of his tests is 8% (or 0.08).

#### a) What is the probability that the class will earn a pizza on the fifth test?
This scenario involves the class not getting an A on the first four tests and then getting an A on the fifth test. This is a classic example of a geometric probability distribution.

The probability can be calculated as:
\[ P(\text{5th test}) = (1 - p)^{4} \times p \]
Where \( p = 0.08 \).

\[
P(\text{5th test}) = (1 - 0.08)^4 \times 0.08 = 0.92^4 \times 0.08
\]

\[
P(\text{5th test}) = 0.7164 \times 0.08 \approx 0.0573
\]

So, the probability is approximately **0.0573** or **5.73%**.

#### b) What is the probability that the class will not earn a pizza for the first seven tests?
This scenario is equivalent to the class failing to get an A on each of the first seven tests. 

The probability can be calculated as:
\[ P(\text{No pizza in 7 tests}) = (1 - p)^7 = 0.92^7 \]

\[
P(\text{No pizza in 7 tests}) \approx 0.5132
\]

So, the probability is approximately **0.5132** or **51.32%**.

#### c) What is the expected waiting time before the class gets a pizza?
In a geometric distribution, the expected number of trials until the first success (in this case, the first time the class earns a pizza) is given by:
\[ E(X) = \frac{1}{p} \]

Where \( p = 0.08 \).

\[
E(X) = \frac{1}{0.08} = 12.5
\]

So, the expected number of tests before the class gets a pizza is **12.5 tests**.

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### Do you have any questions, or would you like more details on any part of this solution?

Here are five related questions to further explore this topic:
1. What is the probability that the class will get a pizza within the first 10 tests?
2. How does the expected waiting time change if the probability of getting an A-average increases to 10%?
3. What is the probability that the class will get a pizza exactly on the 8th test?
4. How would the probability change if the class needs to get two A-averages in a row to earn a pizza?
5. What is the variance of the number of tests until the class gets their first pizza?

**Tip:** The geometric distribution is particularly useful in scenarios where you’re counting the number of trials until the first success occurs. It’s essential to recognize this distribution when dealing with "first occurrence" type problems.

A teacher provides pizza for his class if they earn an A-average on any test. The probability of the class getting an A-average on one of his tests is 8%. What is the probability that the class will earn a pizza on the fifth test? What is the probability that the class will not earn a pizza for the first seven tests? What is the expected waiting time before the class gets a pizza?

Solve this probability problem involving a class earning pizza if they get an A-average. Learn about geometric distribution, calculate the probability of success on the fifth test, and understand the expected waiting time before success. Suitable for high school students.

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