\( P _ { ( 14 ) } = \left( \begin{array} { l } { 26 } \\ { 14 } \end{array} \right) 0,6 ^ { 14 } ( 1 - 0,6 ) ^ { 26 - 14 } \)

The expression you've provided appears to be the probability of getting exactly 14 successes in 26 trials of a binomial experiment, where the probability of success on each trial is 0.6. The formula for the probability of getting exactly \( k \) successes in \( n \) trials in a binomial distribution is given by: \[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \] where: - \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \), - \( p \) is the probability of success on a single trial, - \( n \) is the number of trials, - \( k \) is the number of successes. In your case: - \( n = 26 \) - \( k = 14 \) - \( p = 0.6 \) Now, let's calculate \( P(X = 14) \): 1. Calculate the binomial coefficient \( \binom{26}{14} \): \[ \binom{26}{14} = \frac{26!}{14!(26-14)!} = \frac{26!}{14! \cdot 12!} \] 2. Calculate \( p^k \) and \( (1 - p)^{n - k} \): \[ p^k = 0.6^{14} \] \[ (1 - p)^{n - k} = (0.4)^{12} \] 3. Combine these values into the formula: \[ P(X = 14) = \binom{26}{14} \cdot 0.6^{14} \cdot 0.4^{12} \] Now, let's compute these values step by step. ### Step 1: Calculate the binomial coefficient Using a calculator or software to compute \( \binom{26}{14} \): \[ \binom{26}{14} = 10400600 \] ### Step 2: Calculate \( 0.6^{14} \) and \( 0.4^{12} \) Using a calculator: \[ 0.6^{14} \approx 0.0060466176 \] \[ 0.4^{12} \approx 0.00016777216 \] ### Step 3: Combine everything Now, substitute these values back into the probability formula: \[ P(X = 14) = 10400600 \cdot 0.0060466176 \cdot 0.00016777216 \] Calculating this gives: \[ P(X = 14) \approx 0.104 \] Thus, the probability of getting exactly 14 successes in 26 trials with a success probability of 0.6 is approximately \( 0.104 \) or \( 10.4\% \).