Question: John has $6$ green marbles and $4$ purple marbles. He chooses a marble at random, writes down its color, and then puts the marble back. He performs this process $5$ times. What is the probability that he chooses exactly two green marbles?
1. **State the problem:**
John has $6$ green marbles and $4$ purple marbles, total $6 + 4 = 10$ marbles.
He picks a marble at random, records its color, then replaces it, repeating this $5$ times.
We want the probability of choosing exactly $2$ green marbles in these $5$ trials.
2. **Identify the type of problem:**
This is a binomial probability problem because each trial is independent with two outcomes: green or not green.
3. **Formula for binomial probability:**
$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$
where:
- $n = 5$ (number of trials),
- $k = 2$ (number of successes, green marbles),
- $p$ = probability of success on one trial.
4. **Calculate $p$:**
$$p = \frac{6}{10} = 0.6$$
Probability of not green (purple) is:
$$1-p = 0.4$$
5. **Calculate binomial coefficient:**
$$\binom{5}{2} = \frac{5!}{2! (5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$$
6. **Calculate probability:**
$$P(X=2) = 10 \times (0.6)^2 \times (0.4)^3$$
7. **Intermediate calculation:**
$$(0.6)^2 = 0.36$$
$$(0.4)^3 = 0.064$$
8. **Final calculation:**
$$P(X=2) = 10 \times 0.36 \times 0.064 = 10 \times 0.02304 = 0.2304$$
**Answer:** The probability that John chooses exactly two green marbles in $5$ trials is $0.2304$ or $23.04\%$.