To determine the probability of getting exactly 2 tails when flipping a fair coin 7 times, we can make use of the binomial probability formula. This formula is particularly applicable because we have a fixed number of trials (7 flips), two possible outcomes (heads or tails), and each flip of the coin has the same probability of resulting in heads or tails.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1 – p)^(n – k)
Where:
- P(X = k) = probability of getting exactly k successes (tails in this case)
- C(n, k) = binomial coefficient, which calculates the number of ways to choose k successes from n trials
- p = probability of success on a single trial (for a fair coin, p = 0.5 for tails)
- n = total number of trials (in this case, n = 7)
Now, let’s break it down:
- We want exactly 2 tails (k = 2).
- We have a total of 7 flips (n = 7).
- Since the probability of getting tails in one flip is 0.5, we have p = 0.5.
The binomial coefficient C(7, 2) can be calculated using:
C(n, k) = n! / [k!(n – k)!] = 7! / [2!(7 – 2)!]
Calculating this gives:
C(7, 2) = 7! / [2! * 5!] = (7 * 6) / (2 * 1) = 21
Now, we can calculate the probability:
P(X = 2) = C(7, 2) * (0.5)^2 * (0.5)^(7 – 2)
Which simplifies to:
P(X = 2) = 21 * (0.25) * (0.5)^5
Solving this gives:
P(X = 2) = 21 * (0.25) * (0.03125)
P(X = 2) = 21 * 0.0078125 = 0.1640625
Therefore, the probability of getting exactly 2 tails when flipping a fair coin 7 times is approximately 0.1641, or 16.41%.