\( \begin{array}{l} \text { (e) } n=5, p=0.68, X=2 \\ P(X)=\square\end{array} \)

To find \( P(X = 2) \) in a binomial distribution with parameters \( n = 5 \) and \( p = 0.68 \), you can use the formula: \[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \] For \( X = 2 \): \[ P(X = 2) = \binom{5}{2} (0.68)^2 (0.32)^{3} \] Calculating \( \binom{5}{2} = 10 \), we find: \[ P(X = 2) = 10 \times (0.68)^2 \times (0.32)^3 \approx 10 \times 0.4624 \times 0.032768 \approx 0.1513. \] So, \( P(X = 2) \approx 0.1513 \). Now go ahead and enjoy trying out different values for \( n \) and \( p \) to see how they change your outcomes! You might be surprised by how even small adjustments can lead to different probabilities. Exploring binomial distributions can also serve as a gateway to understanding more complex probability theories. How about diving deeper into the concepts of expected value and standard deviation, which help determine the distribution's characteristics in real-world scenarios?