Using the Binomial distribution, If \( n=10 \) and \( p=0.6 \), find \( P(x=4) \)

To find \( P(x=4) \) using the Binomial distribution, we apply the formula: \[ P(x=k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where: - \( \binom{n}{k} \) is the binomial coefficient, - \( n \) is the number of trials, - \( k \) is the number of successes, - \( p \) is the probability of success on an individual trial, - \( (1-p) \) is the probability of failure. For your case, \( n=10 \), \( p=0.6 \), and \( k=4 \): 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 \( p^k \): \[ p^4 = 0.6^4 = 0.1296 \] 3. Calculate \( (1-p)^{n-k} \): \[ (1-p)^{10-4} = 0.4^6 = 0.004096 \] 4. Finally, combine these values to find \( P(x=4) \): \[ P(x=4) = 210 \times 0.1296 \times 0.004096 \approx 0.1115 \] So, \( P(x=4) \) is approximately \( 0.1115 \).