What is the probability of 5 or fewer successes in a binomial experiment with 7 trials if in each trial, the porbability of success is .8?
To determine the probability of 5 or fewer successes in a binomial experiment with 7 trials, where the probability of success in each trial is 0.8, you can use the cumulative distribution function (CDF) of the binomial distribution.
The formula for the probability mass function (PMF) of the binomial distribution is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
In this formula, n is the number of trials, k is the number of successes, p is the probability of success in a single trial, and (nCk) represents the number of combinations of n items taken k at a time.
To calculate the probability of 5 or fewer successes, you need to compute the sum of the probabilities for k = 0, 1, 2, 3, 4, and 5.
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Now, substitute the values into the formula and calculate:
P(X ≤ 5) = (7C0) * 0.8^0 * (1-0.8)^(7-0) + (7C1) * 0.8^1 * (1-0.8)^(7-1) + (7C2) * 0.8^2 * (1-0.8)^(7-2) + (7C3) * 0.8^3 * (1-0.8)^(7-3) + (7C4) * 0.8^4 * (1-0.8)^(7-4) + (7C5) * 0.8^5 * (1-0.8)^(7-5)
Simplifying the expression by calculating the combinations (nCk), we get:
P(X ≤ 5) = 1 * 0.8^0 * (1-0.8)^7 + 7 * 0.8^1 * (1-0.8)^(7-1) + 21 * 0.8^2 * (1-0.8)^(7-2) + 35 * 0.8^3 * (1-0.8)^(7-3) + 35 * 0.8^4 * (1-0.8)^(7-4) + 21 * 0.8^5 * (1-0.8)^(7-5)
Calculate each term and sum them up to find the probability.