Sure! Let's determine the probability that Claire will get exactly 3 tails when flipping a coin 6 times.
1. Identify the Problem:
We need to find the probability of getting exactly 3 tails in 6 coin flips.
2. Understand the Concept:
When flipping a fair coin, each flip has two possible outcomes: heads (H) or tails (T), each with a probability of [tex]\(0.5\)[/tex]. For a series of flips, certain outcomes can be calculated using the binomial probability formula.
3. Binomial Probability Formula:
The probability of getting exactly [tex]\(k\)[/tex] successes (tails in this case) in [tex]\(n\)[/tex] trials (flips) is given by:
[tex]\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
\][/tex]
where:
- [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, calculated as [tex]\(\frac{n!}{k!(n-k)!}\)[/tex],
- [tex]\(p\)[/tex] is the probability of getting a tail in a single flip (0.5),
- [tex]\(n\)[/tex] is the total number of flips,
- [tex]\(k\)[/tex] is the number of desired tails.
4. Assign Values:
For this problem:
- [tex]\(n = 6\)[/tex],
- [tex]\(k = 3\)[/tex],
- [tex]\(p = 0.5\)[/tex].
5. Calculate the Binomial Coefficient:
The binomial coefficient [tex]\(\binom{6}{3}\)[/tex] represents the number of ways to choose 3 tails from 6 flips.
[tex]\[
\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
\][/tex]
6. Apply the Binomial Probability Formula:
[tex]\[
P(X = 3) = \binom{6}{3} \cdot (0.5)^3 \cdot (0.5)^{6-3}
\][/tex]
[tex]\[
P(X = 3) = 20 \cdot (0.5)^3 \cdot (0.5)^3
\][/tex]
[tex]\[
P(X = 3) = 20 \cdot (0.5)^6
\][/tex]
[tex]\[
P(X = 3) = 20 \cdot 0.015625
\][/tex]
[tex]\[
P(X = 3) = 0.3125
\][/tex]
Therefore, the probability that Claire will get exactly 3 tails when flipping a coin 6 times is [tex]\(\boxed{0.313}\)[/tex].
