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The problem provided is about finding the probability that exactly 4 heads come up when flipping a biased coin 11 times, where the probability of heads is \(0.2\). This is a **binomial probability** problem.

The binomial probability formula is:

\[
P(k = 4) = \binom{n}{k} p^k (1 - p)^{n - k}
\]

Where:
- \(n = 11\) (the number of trials, or flips),
- \(k = 4\) (the number of successful outcomes, or heads),
- \(p = 0.2\) (the probability of success, or heads on a single flip),
- \(1 - p = 0.8\) (the probability of failure, or tails on a single flip),
- \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) is the binomial coefficient.

We will calculate this step-by-step:

### Step 1: Compute the binomial coefficient
\[
\binom{11}{4} = \frac{11!}{4!(11-4)!} = \frac{11!}{4!7!}
\]

### Step 2: Apply the binomial formula
\[
P(k = 4) = \binom{11}{4} (0.2)^4 (0.8)^7
\]

I'll calculate this now.The probability that exactly 4 flips come up heads when a biased coin (with a probability of heads \(0.2\)) is flipped 11 times is approximately **0.1107**.

Would you like further details or clarifications? 

Here are five related questions to expand your understanding:
1. How would the probability change if the coin were flipped 15 times instead of 11?
2. What is the expected number of heads in 11 flips of the biased coin?
3. How would the probability change if we wanted 5 heads instead of 4?
4. Can you calculate the probability of getting no heads (0 heads) in 11 flips?
5. How does the binomial distribution differ from other discrete probability distributions?

**Tip**: The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success.

The probability that a biased coin comes up heads is 0.2. Let k be the number of flips that come up heads when the coin is flipped 11 times. What is the probability that 4 flips come up heads?

Learn how to calculate the probability of getting exactly 4 heads in 11 flips of a biased coin where the probability of heads is 0.2. This problem demonstrates the use of the binomial probability formula, including detailed calculations and insights.

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