Find the binominal probability...n = 15, p = .04, find P(fewer than 3)
To find the binomial probability, we can use the formula:
P(x) = ( n C x ) * p^x * (1 - p)^(n - x)
Where:
P(x) is the probability of exactly x successful outcomes,
n is the total number of trials,
p is the probability of success in a single trial,
( n C x ) is the binomial coefficient, also known as "n choose x," which represents the number of ways to choose x successes from n trials.
To find P(fewer than 3), we need to calculate P(0) + P(1) + P(2). Let's break it down step by step:
1. Calculate P(0):
P(0) = ( 15 C 0 ) * 0.04^0 * (1 - 0.04)^(15 - 0)
= 1 * 1 * (0.96)^15
≈ 0.4858
2. Calculate P(1):
P(1) = ( 15 C 1 ) * 0.04^1 * (1 - 0.04)^(15 - 1)
= 15 * 0.04^1 * 0.96^14
≈ 0.3942
3. Calculate P(2):
P(2) = ( 15 C 2 ) * 0.04^2 * (1 - 0.04)^(15 - 2)
= (15 * 14 / 2) * 0.04^2 * 0.96^13
≈ 0.1516
Finally, we sum up the probabilities:
P(fewer than 3) = P(0) + P(1) + P(2)
= 0.4858 + 0.3942 + 0.1516
≈ 1.0316
Note that it's unusual for probabilities to sum up to more than 1. In this case, it could be due to rounding errors, as the individual probabilities were rounded at each step. In practice, probabilities should always be between 0 and 1.