For a binomial distribution X~Bin(n,p), suppose n=5. If P(X=0)=0.00032, what is p? In your response, provide a detailed explanation of your steps and reasoning at each step.

We are given a binomial distribution \( X \sim \text{Bin}(n, p) \), with \( n = 5 \), and we need to find \( p \), the probability of success on each trial, given that \( P(X = 0) = 0.00032 \).

### Step 1: The binomial probability mass function (PMF)
The probability mass function for a binomial distribution is given by:

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

Where:
- \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes from \( n \) trials.
- \( p \) is the probability of success.
- \( (1 - p) \) is the probability of failure.
- \( n \) is the number of trials, and \( k \) is the number of successes.

In this case, we want to find \( P(X = 0) \), which means no successes occur in 5 trials.

### Step 2: Applying the binomial formula for \( P(X = 0) \)
For \( X = 0 \), the formula becomes:

\[
P(X = 0) = \binom{5}{0} p^0 (1 - p)^5 = (1 - p)^5
\]

Since \( \binom{5}{0} = 1 \) and \( p^0 = 1 \), the equation simplifies to:

\[
P(X = 0) = (1 - p)^5
\]

### Step 3: Solving for \( p \)
We are given \( P(X = 0) = 0.00032 \). So, we set up the equation:

\[
(1 - p)^5 = 0.00032
\]

Now, to solve for \( p \), we take the 5th root of both sides:

\[
1 - p = \sqrt[5]{0.00032}
\]

Using a calculator:

\[
\sqrt[5]{0.00032} \approx 0.2
\]

Thus, \( 1 - p = 0.2 \).

### Step 4: Isolating \( p \)
Now, solve for \( p \):

\[
p = 1 - 0.2 = 0.8
\]

### Final Answer:
The probability of success, \( p \), is \( 0.8 \).

### Recap of the steps:
1. We used the binomial PMF formula and applied it for \( P(X = 0) \).
2. Simplified the expression for \( P(X = 0) \).
3. Set the equation equal to 0.00032 and took the 5th root.
4. Solved for \( p \).

Would you like more details or clarifications on any step?

### Related Questions:
1. What is the probability of getting exactly 2 successes with \( n = 5 \) and \( p = 0.8 \)?
2. How would the probability change if \( n \) were increased to 10 trials?
3. How can we use the cumulative distribution function for binomial distributions?
4. What is the variance of a binomial distribution with \( n = 5 \) and \( p = 0.8 \)?
5. How can we apply the normal approximation to the binomial distribution?

### Tip:
For a binomial distribution, the mean is \( \mu = np \) and the variance is \( \sigma^2 = np(1 - p) \). These are useful for approximations and understanding the distribution's behavior.

Discover how to find the probability of success p in a binomial distribution X~Bin(5, p) given P(X=0)=0.00032. This problem covers key concepts in probability theory, such as the binomial distribution and binomial probability formula. Suitable for high school students in Grades 11-12.

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