Step 1 :First, we need to calculate the mean and standard deviation of the binomial distribution. The mean (expected value) of a binomial distribution is given by \(\mu = np\), and the standard deviation is given by \(\sigma = \sqrt{np(1-p)}\).
Step 2 :Substitute \(n = 50\) and \(p = 0.1\) into the formulas. We get \(\mu = 50 \times 0.1 = 5\) and \(\sigma = \sqrt{50 \times 0.1 \times (1-0.1)} = \sqrt{4.5} \approx 2.12\).
Step 3 :Next, we need to standardize the random variable \(X = 15\) using the formula \(Z = \frac{X - \mu}{\sigma}\).
Step 4 :Substitute \(X = 15\), \(\mu = 5\), and \(\sigma = 2.12\) into the formula. We get \(Z = \frac{15 - 5}{2.12} \approx 4.72\).
Step 5 :Now, we can use the standard normal distribution table to find the probability \(P(Z < 4.72)\). However, the standard normal distribution table usually does not provide values for \(Z > 3.49\), because the probabilities are very close to 1.
Step 6 :Therefore, we can say that \(P(Z < 4.72) \approx 1\).
Step 7 :Finally, because \(X = 15\) is greater than the mean \(\mu = 5\), we need to subtract this probability from 1 to get the probability \(P(X = 15)\).
Step 8 :So, \(P(X = 15) = 1 - P(Z < 4.72) = 1 - 1 = 0\).
Step 9 :Therefore, the probability that \(X = 15\) is approximately 0 when using the normal distribution to approximate the binomial distribution.