Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places.
P(X≤4)
, n=6
, p=0.4
To find P(X ≤ 4) for a binomial distribution with n = 6 and p = 0.4, we need to calculate the individual probabilities of getting 0, 1, 2, 3, or 4 successes.
Using the formula for the probability mass function of a binomial distribution:
P(X = k) = (n choose k) * p^k * (1 - p)^(n-k)
where (n choose k) represents the number of ways to choose k successes out of n trials, we can calculate the probabilities for k = 0, 1, 2, 3, 4.
P(X = 0) = (6 choose 0) * 0.4^0 * 0.6^6 ≈ 0.0467
P(X = 1) = (6 choose 1) * 0.4^1 * 0.6^5 ≈ 0.1867
P(X = 2) = (6 choose 2) * 0.4^2 * 0.6^4 ≈ 0.3110
P(X = 3) = (6 choose 3) * 0.4^3 * 0.6^3 ≈ 0.2765
P(X = 4) = (6 choose 4) * 0.4^4 * 0.6^2 ≈ 0.1382
To find P(X ≤ 4), we sum up the probabilities of getting 0, 1, 2, 3, or 4 successes:
P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) ≈ 0.9591
Therefore, P(X ≤ 4) for a binomial distribution with n = 6 and p = 0.4 is approximately 0.9591.