To find the probability of getting exactly 6 heads when flipping a fair coin 9 times, we can use the binomial probability formula. The formula for calculating the probability of getting exactly k successes (heads) in n trials (flips) is:
P(X = k) = C(n, k) * p^k * (1 – p)^(n – k)
Where:
- C(n, k) is the number of combinations of n items taken k at a time, calculated as C(n, k) = n! / (k!(n – k)!).
- p is the probability of success on an individual trial (for a fair coin, p = 0.5).
- n is the total number of trials (in this case, 9 flips).
- k is the number of successful trials we are interested in (in this case, 6 heads).
Let’s plug in the numbers:
– n = 9
– k = 6
– p = 0.5
Now, we first calculate C(9, 6):
C(9, 6) = 9! / (6! * (9 – 6)!) = 9! / (6! * 3!) = (9 * 8 * 7) / (3 * 2 * 1) = 84
Next, we calculate the probability:
P(X = 6) = C(9, 6) * (0.5)^6 * (0.5)^(9 – 6) = 84 * (0.5)^6 * (0.5)^3 = 84 * (0.5)^9 = 84 / 512 = 0.1640625
Therefore, the probability of getting exactly 6 heads when flipping a fair coin 9 times is approximately 0.164 or 16.4%.