Find p of the Binomial distribution if n = 6 and P(X = 4) = P(X = 2).

Given that \( X \) follows a binomial distribution with parameters \( n = 6 \) and \( p \) (probability of success), and we are given that \( P(X = 4) = P(X = 2) \).

The probability mass function for a binomial distribution is given by:

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

For \( P(X = 4) = P(X = 2) \), we have:

\[
\binom{6}{4} p^4 (1-p)^2 = \binom{6}{2} p^2 (1-p)^4
\]

### Step 1: Simplify the equation
First, compute the binomial coefficients:

\[
\binom{6}{4} = \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15
\]

Substitute these values into the equation:

\[
15 p^4 (1-p)^2 = 15 p^2 (1-p)^4
\]

Since \( 15 \) appears on both sides of the equation, we can cancel it out:

\[
p^4 (1-p)^2 = p^2 (1-p)^4
\]

### Step 2: Factor and solve for \( p \)

We can divide both sides by \( p^2(1-p)^2 \) (assuming \( p \neq 0 \) and \( p \neq 1 \)):

\[
p^2 = (1-p)^2
\]

Taking the square root of both sides:

\[
p = 1 - p
\]

Solving for \( p \):

\[
2p = 1 \quad \Rightarrow \quad p = \frac{1}{2}
\]

### Final Answer:
The probability of success \( p \) is \( \frac{1}{2} \).

Do you need further clarification on any steps?

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Here are five related questions to consider:

1. What would the probability \( P(X = 3) \) be if \( p = \frac{1}{2} \) and \( n = 6 \)?
2. How would the probabilities change if \( n = 10 \) and \( p = 0.5 \)?
3. What is the expected value \( E(X) \) for this binomial distribution?
4. How do the symmetry properties of the binomial distribution help in finding \( p \) when \( P(X = k) = P(X = n-k) \)?
5. What is the variance of the distribution when \( p = \frac{1}{2} \) and \( n = 6 \)?

**Tip:** In a binomial distribution with \( p = \frac{1}{2} \), the distribution is symmetric around the mean, making it easier to find probabilities for complementary events.

Explore how to find the probability of success (p) in a binomial distribution with n = 6, given P(X = 4) = P(X = 2). This problem involves using the binomial probability mass function and solving for p with step-by-step calculations.

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