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.



In this problem, we are given that \( X \) follows a binomial distribution with parameters \( n = 5 \) and an unknown probability of success \( p \), and we need to determine the value of \( p \) given that \( P(X = 0) = 0.00032 \).

### Step 1: The binomial distribution formula

The probability mass function (PMF) for a binomial distribution \( X \sim \text{Bin}(n, p) \) is given by the formula:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
where:
- \( n \) is the number of trials,
- \( p \) is the probability of success on a single trial,
- \( k \) is the number of successes,
- \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes from \( n \) trials.

In our case, we are interested in \( P(X = 0) \), which is the probability of having zero successes.

### Step 2: Apply the formula for \( P(X = 0) \)

Using the binomial distribution formula for \( P(X = 0) \):
\[
P(X = 0) = \binom{5}{0} p^0 (1-p)^{5}
\]
Simplifying the terms:
\[
P(X = 0) = 1 \cdot (1-p)^{5}
\]
So, the probability of getting zero successes is simply:
\[
P(X = 0) = (1-p)^{5}
\]

### Step 3: Set up the equation

We are given that \( P(X = 0) = 0.00032 \), so we can set up the equation:
\[
(1-p)^{5} = 0.00032
\]

### Step 4: Solve for \( p \)

To solve for \( p \), first take the fifth root of both sides of the equation:
\[
1 - p = \sqrt[5]{0.00032}
\]
Now, calculate the fifth root of 0.00032. We can do this by raising 0.00032 to the power of \( \frac{1}{5} \):
\[
1 - p = (0.00032)^{\frac{1}{5}}
\]

We compute the value of \( (0.00032)^{\frac{1}{5}} \).

\[
1 - p \approx 0.2
\]

### Step 5: Solve for \( p \)

Now, solve for \( p \) by subtracting 0.2 from 1:
\[
p = 1 - 0.2
\]
\[
p = 0.8
\]

### Final Answer:

Thus, the value of \( p \) is \( \boxed{0.8} \).

---

Would you like more details or have any questions about the steps?

Here are 5 related questions for practice:
1. What is \( P(X = 1) \) for this binomial distribution with \( p = 0.8 \)?
2. How would you calculate the expected value of \( X \) for \( X \sim \text{Bin}(5, 0.8) \)?
3. What is the probability that \( X \geq 3 \) for this binomial distribution?
4. What is the variance of the distribution \( X \sim \text{Bin}(5, 0.8) \)?
5. How does the binomial distribution change as \( n \) increases for a fixed \( p \)?

**Tip**: When solving for unknowns in probability equations, taking roots is often necessary—make sure to review your exponent rules!

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.

Learn how to find the probability of success (p) in a binomial distribution where n=5 and P(X=0)=0.00032. This detailed step-by-step solution covers key probability concepts and the binomial distribution formula. Suitable for Grades 10-12.

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