To find the probability of getting exactly 3 tails when flipping a fair coin 5 times, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 – p)^(n – k)
Where:
- n = number of trials (in this case, 5 flips)
- k = number of successful trials (we want 3 tails)
- p = probability of success on each trial (for a fair coin, p = 0.5 for tails)
- C(n, k) = the binomial coefficient, which is the number of ways to choose k successes from n trials.
First, we calculate the binomial coefficient C(5, 3), which represents the different ways to get 3 tails in 5 flips:
C(5, 3) = 5! / (3! * (5-3)!) = 10
Now, substituting values into the formula:
P(X = 3) = C(5, 3) * (0.5)^3 * (0.5)^(5-3)
This simplifies to:
P(X = 3) = 10 * (0.5)^3 * (0.5)^2 = 10 * (0.5)^5 = 10 * (1/32) = 10/32 = 5/16
Thus, the probability of getting exactly 3 tails when flipping a fair coin 5 times is 5/16 or about 0.3125 (31.25%).