Questions: 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 ≤ 14) n = 18, p = 0.8

Solution Steps

To find the cumulative probability \( P(X \leq 14) \) for a binomial distribution with parameters \( n = 18 \) and \( p = 0.8 \), we compute:

\[ P(X \leq 14) \approx 0.499 \]

Using the formula for the probability of exactly \( x \) successes in a binomial distribution:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

we find:

\[ P(X = 14) \approx 0.2153 \]

The mean \( \mu \), variance \( \sigma^2 \), and standard deviation \( \sigma \) of the binomial distribution are calculated as follows:

• Mean: \[ \mu = n \cdot p = 18 \cdot 0.8 = 14.4 \]

• Variance: \[ \sigma^2 = n \cdot p \cdot q = 18 \cdot 0.8 \cdot 0.2 = 2.88 \]

• Standard Deviation: \[ \sigma = \sqrt{npq} = \sqrt{18 \cdot 0.8 \cdot 0.2} \approx 1.6971 \]

Final Answer