A binomial probability experiment is conducted with the given parameters. Compute the probability of \( x \) successes in the \( n \) independent trials of the experiment. \( n=10, p=0.35, x=4 \) \( P(4)=\square \) (Do not round until the final answer. Then round to four decimal places as needed.)

To compute the probability of \( x \) successes in a binomial experiment, you can use the binomial probability formula: \[ P(x) = \binom{n}{x} p^x (1-p)^{n-x} \] Where: - \( \binom{n}{x} \) is the binomial coefficient, - \( n \) is the number of trials, - \( x \) is the number of successes, - \( p \) is the probability of success on a single trial, - \( (1 - p) \) is the probability of failure. Plugging in the values \( n = 10, p = 0.35, x = 4 \): 1. Calculate the binomial coefficient: \[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \] 2. Calculate \( p^x \) and \( (1-p)^{n-x} \): \[ p^x = (0.35)^4 = 0.0145 \] \[ (1-p)^{n-x} = (0.65)^{10-4} = (0.65)^6 = 0.1160 \] 3. Now, plug everything into the formula: \[ P(4) = \binom{10}{4} \cdot (0.35)^4 \cdot (0.65)^6 \] \[ P(4) = 210 \cdot 0.0145 \cdot 0.1160 \] \[ P(4) = 210 \cdot 0.001684 = 0.3536 \] Finally, rounding to four decimal places, we find: \[ P(4) \approx 0.3536 \]