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Use what we learned about sequential events in module 12 to state the product used to find the value in the probability distribution table for 10.  In other words, the probability that all 10 students succeed in Math 247.

The given formula represents a binomial probability distribution function:

\[
P(x) = \binom{n}{x} \cdot (p)^x \cdot (1 - p)^{n - x}
\]

Where:
- \( \binom{n}{x} \) is the binomial coefficient, representing the number of ways to choose \(x\) successes out of \(n\) trials.
- \( p \) is the probability of success on a single trial.
- \( 1 - p \) is the probability of failure on a single trial.
- \( n \) is the total number of trials.
- \( x \) is the number of successes.

In your case, the formula is:
\[
P(x) = 10Cx \cdot (0.63)^x \cdot (0.37)^{10 - x}
\]
This implies:
- \( n = 10 \) (representing the total number of students)
- \( p = 0.63 \) (probability of success for each student in Math 247)
- \( 1 - p = 0.37 \) (probability of failure)

Now, the problem asks for the probability that **all 10 students succeed** in Math 247. This implies \( x = 10 \).

To find the probability for \( x = 10 \), substitute into the formula:

\[
P(10) = \binom{10}{10} \cdot (0.63)^{10} \cdot (0.37)^{0}
\]

Since \( \binom{10}{10} = 1 \) and \( (0.37)^{0} = 1 \), the expression simplifies to:

\[
P(10) = 1 \cdot (0.63)^{10} \cdot 1 = (0.63)^{10}
\]

Now, calculate \( (0.63)^{10} \):

\[
P(10) \approx (0.63)^{10} \approx 0.01698
\]

Thus, the probability that all 10 students succeed in Math 247 is approximately **0.01698**.

Do you have any further questions or would you like more details?

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Here are five follow-up questions:
1. How would the probability change if the success rate were 0.7 instead of 0.63?
2. What is the probability that exactly 8 out of 10 students succeed?
3. How does the binomial coefficient \( \binom{n}{x} \) influence the overall probability?
4. How would you calculate the probability that fewer than 5 students succeed?
5. What happens to the distribution when the number of trials \( n \) increases to 20?

**Tip**: Binomial probability distributions are useful when you have a fixed number of independent trials with two possible outcomes (success or failure), and the probability of success remains constant across all trials.

Use what we learned about sequential events in module 12 to state the product used to find the value in the probability distribution table for 10. In other words, the probability that all 10 students succeed in Math 247.

This problem explores the binomial probability distribution to calculate the probability that all 10 students succeed in Math 247, given a 63% success rate. Learn how the binomial coefficient and success probabilities are applied to find the result.

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