A binomial probability experiment is conducted with the given parameters Compute the probability of x successes in the n independent trials of the experim n=7.p=0.4,x<=4

1. The probability mass function for a binomial distribution is given by:

P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}

2. For each \( x \) from \( 0 \) to \( 4 \), calculate the individual probabilities:

P(X = 0) = \binom{7}{0} (0.4)^0 (0.6)^7 = 0.2799

P(X = 1) = \binom{7}{1} (0.4)^1 (0.6)^6 = 0.3917

P(X = 2) = \binom{7}{2} (0.4)^2 (0.6)^5 = 0.2646

P(X = 3) = \binom{7}{3} (0.4)^3 (0.6)^4 = 0.1176

P(X = 4) = \binom{7}{4} (0.4)^4 (0.6)^3 = 0.0375

3. Add the probabilities from \( x = 0 \) to \( x = 4 \):

P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

P(X \leq 4) = 0.2799 + 0.3917 + 0.2646 + 0.1176 + 0.0375 \approx 0.9013

4. Therefore, the probability of \( x \leq 4 \) successes is:

P(X \leq 4) \approx 0.9013