A binomial experiment has the given number of trials \( n \) and the given success probability \( p \). \[ n=13, p=0.4 \] Part: \( \mathbf{0} / \mathbf{3} \) Part 1 of 3 (a) Determine the probability \( P \) (Fewer than 3). Round the answer to at least three decimal places. \( P( \) Fewer than 3\( )= \)

Given a binomial experiment with \[ n = 13,\quad p = 0.4,\quad q = 1-p = 0.6, \] we want to determine the probability of fewer than 3 successes, i.e., \[ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2). \] The probability mass function for a binomial distribution is \[ P(X=k) = \binom{n}{k} p^k q^{n-k}. \] **Step 1. Compute \(P(X=0)\):** \[ P(X=0) = \binom{13}{0} (0.4)^0 (0.6)^{13} = 1 \times 1 \times (0.6)^{13} = (0.6)^{13}. \] Calculating \(0.6^{13}\) (approximately): \[ 0.6^{13} \approx 0.00131. \] **Step 2. Compute \(P(X=1)\):** \[ P(X=1) = \binom{13}{1} (0.4)^1 (0.6)^{12} = 13 \times 0.4 \times (0.6)^{12}. \] First, compute \(13 \times 0.4\): \[ 13 \times 0.4 = 5.2. \] Now, using the approximate value \[ 0.6^{12} \approx 0.00218, \] we have \[ P(X=1) \approx 5.2 \times 0.00218 \approx 0.01133. \] **Step 3. Compute \(P(X=2)\):** \[ P(X=2) = \binom{13}{2} (0.4)^2 (0.6)^{11}. \] First, calculate the binomial coefficient: \[ \binom{13}{2} = \frac{13 \times 12}{2} = 78. \] Then, \[ (0.4)^2 = 0.16. \] Multiplying these gives: \[ 78 \times 0.16 = 12.48. \] Using the approximate value \[ 0.6^{11} \approx 0.00363, \] we find \[ P(X=2) \approx 12.48 \times 0.00363 \approx 0.04524. \] **Step 4. Sum the Probabilities:** \[ P(X<3) = P(X=0) + P(X=1) + P(X=2) \approx 0.00131 + 0.01133 + 0.04524 = 0.05788. \] Rounded to at least three decimal places, \[ P(X<3) \approx 0.058. \] Thus, \[ P(\text{Fewer than } 3) \approx 0.058. \]