To determine the probability of getting exactly 4 tails when a coin is flipped 10 times, we can use the binomial probability formula. This is because the situation described follows a binomial distribution, where there are a fixed number of independent trials (coin flips), each with two possible outcomes (heads or tails), and a constant probability of success (getting tails) on each trial.
Here's how we can solve it step-by-step:
1. Identify the parameters of the binomial distribution:
- Number of trials ([tex]\(n\)[/tex]): This is the number of times the coin is flipped, which is 10.
- Number of successes ([tex]\(k\)[/tex]): This is the number of times we want to get tails, which is 4.
- Probability of success on a single trial ([tex]\(p\)[/tex]): This is the probability of getting tails in a single flip of the coin, which is 0.5 since the coin is fair.
2. Binomial Probability Formula:
The probability of getting exactly [tex]\(k\)[/tex] successes (tails) in [tex]\(n\)[/tex] independent Bernoulli trials (coin flips) is given by the binomial probability formula:
[tex]\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\][/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 success on a single trial,
[tex]\(1-p\)[/tex] is the probability of failure on a single trial.
3. Apply the Values to the Formula:
Plugging in the values:
[tex]\[
P(X = 4) = \binom{10}{4} (0.5)^4 (0.5)^{10-4}
\][/tex]
4. Calculate the Binomial Coefficient:
[tex]\[
\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210
\][/tex]
5. Compute the Remaining Parts:
Since [tex]\( (0.5)^4 \)[/tex] and [tex]\( (0.5)^6 \)[/tex]:
[tex]\[
P(X = 4) = 210 \times (0.5)^4 \times (0.5)^6
\][/tex]
Simplify the exponents:
[tex]\[
P(X = 4) = 210 \times (0.5)^{4+6} = 210 \times (0.5)^{10} = 210 \times \frac{1}{1024}
\][/tex]
6. Final Calculation:
[tex]\[
P(X = 4) = \frac{210}{1024} \approx 0.205078125
\][/tex]
Therefore, the probability of getting exactly 4 tails when flipping a coin 10 times is approximately [tex]\(0.205078125\)[/tex] or about [tex]\(20.5\%\)[/tex].
