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 $\boxed{0.8}$.

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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!