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=0.6
To find P(X<4), we need to calculate the probability of getting 0, 1, 2, or 3 successes.
P(X=0) = (6 choose 0) * (0.6)^0 * (0.4)^(6-0) = 1 * 1 * 0.4^6 = 0.0041
P(X=1) = (6 choose 1) * (0.6)^1 * (0.4)^(6-1) = 6 * 0.6 * 0.4^5 = 0.0467
P(X=2) = (6 choose 2) * (0.6)^2 * (0.4)^(6-2) = 15 * 0.36 * 0.4^4 = 0.1866
P(X=3) = (6 choose 3) * (0.6)^3 * (0.4)^(6-3) = 20 * 0.216 * 0.4^3 = 0.3110
Adding these probabilities together:
P(X<4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = 0.0041 + 0.0467 + 0.1866 + 0.3110 = 0.5484
Therefore, P(X<4) when n=6 and p=0.6 is approximately 0.5484.