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.

Given that \( X \sim \text{Bin}(n, p) \), where \( X \) is a binomially distributed random variable with parameters \( n = 5 \) and \( p \) (the probability of success in a single trial), and that \( P(X = 0) = 0.00032 \), we need to find \( p \).

### Step-by-Step Solution:

1. **Binomial Distribution Formula**:
   The probability mass function (PMF) of a binomial distribution is given by:

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

   Here, \( n = 5 \), and we are given \( P(X = 0) = 0.00032 \), so we substitute \( k = 0 \) into the formula:

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

2. **Simplifying the Formula**:
   Recall that \( \binom{5}{0} = 1 \) and \( p^0 = 1 \), so the equation simplifies to:

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

   We are given that \( P(X = 0) = 0.00032 \), so:

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

3. **Solving for \( p \)**:
   To solve for \( p \), we first take the fifth root of both sides:

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

   Now, calculate \( \sqrt[5]{0.00032} \).

   \[
   1 - p = 0.2
   \]

4. **Final Calculation**:
   Solving for \( p \):

   \[
   p = 1 - 0.2 = 0.8
   \]

Thus, the probability \( p \) is **0.8**.

### Explanation:

- We used the binomial distribution formula, focusing on the case where \( X = 0 \).
- After simplifying the binomial coefficient and powers, we reduced the problem to solving the equation \( (1 - p)^5 = 0.00032 \).
- Taking the fifth root allowed us to isolate \( p \), leading to the final solution \( p = 0.8 \).

Would you like more details or have any questions?

### Related Questions:
1. How would the answer change if \( P(X = 1) \) were provided instead of \( P(X = 0) \)?
2. What is the expected value \( E(X) \) for this binomial distribution?
3. What is the variance of the binomial distribution \( X \sim \text{Bin}(5, 0.8) \)?
4. How do you compute the cumulative distribution function (CDF) for a binomial distribution?
5. What happens to the distribution as \( n \) increases but \( p \) remains constant?

### Tip:
When solving binomial distribution problems, always ensure that you simplify factorial and binomial coefficient terms early to avoid unnecessary complexity.

Learn how to calculate the probability p in a binomial distribution X~Bin(n, p) with n=5 and P(X=0)=0.00032. This detailed step-by-step solution covers the binomial distribution formula and key probability concepts. Suitable for high school students in Grades 10-12.

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