Put in Binomial Probability Formula? probability ( 2 of 22 are lost)
= C(22,2) (1/200)^2 (199/200)^20
The binomial probability formula is used to calculate the probability of a specific number of successes in a fixed number of independent trials, where each trial has the same probability of success.
In this case, you are calculating the probability that exactly 2 out of 22 items are lost.
The formula for binomial probability is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of exactly k successes,
C(n, k) is the number of combinations of n items taken k at a time (also known as n choose k),
p is the probability of a single success in one trial, and
n is the total number of trials.
For your specific question, the formula would be:
P(X = 2) = C(22, 2) * (1/200)^2 * (199/200)^(22-2)
Let's calculate it step by step:
1. Calculate C(22, 2):
C(22, 2) = 22! / (2! * (22 - 2)!)
= 22! / (2! * 20!)
= (22 * 21) / (2 * 1)
= 231
2. Calculate (1/200)^2:
(1/200)^2 = 1 / (200^2)
= 1 / 40,000
= 0.000025
3. Calculate (199/200)^(22-2):
(199/200)^(20) = (199/200)^20
≈ 0.9048
4. Substitute the calculated values into the formula:
P(X = 2) ≈ C(22,2) * (1/200)^2 * (199/200)^20
≈ 231 * 0.000025 * 0.9048
≈ 0.0528
Therefore, the probability that exactly 2 out of 22 items are lost is approximately 0.0528, or 5.28%.