Answer :
To determine the probability of getting at most 4 successes (\(x \leq 4\)) in a binomial experiment with parameters \(n = 13\) and \(p = 0.4\), follow these steps:
1. Understand the parameters:
- \(n\): The number of trials is 13.
- \(p\): The probability of success on each trial is 0.4.
- \(x \leq 4\): We need the probability of getting at most 4 successes.
2. Use the cumulative distribution function (CDF) of the binomial distribution to find the cumulative probability \(P(X \leq 4)\). The CDF for a binomial distribution \(B(n, p)\) sums up the probabilities of getting 0, 1, 2, 3, and 4 successes.
3. Sum the probabilities for each value from \(x = 0\) to \(x = 4\):
[tex]\[ P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) \][/tex]
4. Calculate individual probabilities using the binomial probability formula:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
where \(\binom{n}{k}\) is the binomial coefficient.
5. Sum the calculated probabilities to get cumulative probability for \(X \leq 4\).
After performing all the above calculations, we would find that the cumulative probability for \(x \leq 4\) successes in 13 trials with success probability 0.4 is approximately 0.353.
Therefore, the probability of [tex]\(x \leq 4\)[/tex] successes is [tex]\(\boxed{0.353}\)[/tex], rounded to four decimal places.
1. Understand the parameters:
- \(n\): The number of trials is 13.
- \(p\): The probability of success on each trial is 0.4.
- \(x \leq 4\): We need the probability of getting at most 4 successes.
2. Use the cumulative distribution function (CDF) of the binomial distribution to find the cumulative probability \(P(X \leq 4)\). The CDF for a binomial distribution \(B(n, p)\) sums up the probabilities of getting 0, 1, 2, 3, and 4 successes.
3. Sum the probabilities for each value from \(x = 0\) to \(x = 4\):
[tex]\[ P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) \][/tex]
4. Calculate individual probabilities using the binomial probability formula:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
where \(\binom{n}{k}\) is the binomial coefficient.
5. Sum the calculated probabilities to get cumulative probability for \(X \leq 4\).
After performing all the above calculations, we would find that the cumulative probability for \(x \leq 4\) successes in 13 trials with success probability 0.4 is approximately 0.353.
Therefore, the probability of [tex]\(x \leq 4\)[/tex] successes is [tex]\(\boxed{0.353}\)[/tex], rounded to four decimal places.
