A binomial probability experiment is conducted with the given parameters. Compute the probability of \( x \) successes in the \( n \) independent trials of the experiment. \( n=12, p=0.35, x=2 \) \( P(2)=\square \) (Do not round untit the final answer. Then round to four decimal places as needed.)

To compute the binomial probability \( P(x) \) in a binomial experiment, we can use the formula: \[ P(x) = \binom{n}{x} p^x (1-p)^{n-x} \] Where: - \( n \) is the number of trials (12 in this case), - \( x \) is the number of successes (2 here), - \( p \) is the probability of success (0.35), - \( \binom{n}{x} \) is the binomial coefficient, calculated as \( \frac{n!}{x!(n-x)!} \). 1. Calculate the binomial coefficient: \[ \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \] 2. Calculate \( p^x \): \[ (0.35)^2 = 0.1225 \] 3. Calculate \( (1-p)^{n-x} \): \[ (1-0.35)^{12-2} = (0.65)^{10} \approx 0.0133 \] 4. Now, combine all parts: \[ P(2) = \binom{12}{2} \cdot (0.35)^2 \cdot (0.65)^{10} \approx 66 \cdot 0.1225 \cdot 0.0133 \approx 0.11522 \] Finally, rounding to four decimal places: \[ P(2) \approx 0.1152 \] So, the probability \( P(2) \) is approximately 0.1152.