Question 1 Given the number of trials and the probability of success, determine the probability indicated: \( \mathrm{n}=12, \mathrm{p}=0.7 \), find \( \mathrm{P}(7 \) successes) \( \mathrm{P}(7 \) successes \( )= \) Submit Question

To find the probability of 7 successes in 12 trials with a probability of success of 0.7, we can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where: - \( n \) is the number of trials (12 in this case), - \( k \) is the number of successes (7 in this case), - \( p \) is the probability of success (0.7 in this case), - \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes out of \( n \) trials. Substitute the given values into the formula: \[ P(7 \text{ successes}) = \binom{12}{7} (0.7)^7 (1-0.7)^{12-7} \] Now, we can calculate the probability using the binomial probability formula. Calculate the value by following steps: - step0: Calculate: \( { }_{12}C_{7}\times 0.7^{7}\left(1-0.7\right)^{12-7}\) - step1: Subtract the numbers: \( { }_{12}C_{7}\times 0.7^{7}\times 0.3^{12-7}\) - step2: Subtract the numbers: \( { }_{12}C_{7}\times 0.7^{7}\times 0.3^{5}\) - step3: Expand the expression: \(\frac{12!}{7!\times \left(12-7\right)!}\times 0.7^{7}\times 0.3^{5}\) - step4: Subtract the numbers: \(\frac{12!}{7!\times 5!}\times 0.7^{7}\times 0.3^{5}\) - step5: Reduce the fraction: \(\frac{12\times 11\times 10\times 9\times 8}{5!}\times 0.7^{7}\times 0.3^{5}\) - step6: Multiply: \(\frac{95040}{5!}\times 0.7^{7}\times 0.3^{5}\) - step7: Divide the terms: \(792\times 0.7^{7}\times 0.3^{5}\) - step8: Convert the expressions: \(792\left(\frac{7}{10}\right)^{7}\times 0.3^{5}\) - step9: Convert the expressions: \(792\left(\frac{7}{10}\right)^{7}\left(\frac{3}{10}\right)^{5}\) - step10: Simplify: \(792\times \frac{7^{7}}{10^{7}}\times \left(\frac{3}{10}\right)^{5}\) - step11: Multiply the terms: \(\frac{99\times 7^{7}}{16\times 5^{7}}\times \left(\frac{3}{10}\right)^{5}\) - step12: Evaluate the power: \(\frac{99\times 7^{7}}{16\times 5^{7}}\times \frac{3^{5}}{10^{5}}\) - step13: Multiply the fractions: \(\frac{99\times 7^{7}\times 3^{5}}{16\times 5^{7}\times 10^{5}}\) - step14: Multiply: \(\frac{24057\times 7^{7}}{16\times 5^{7}\times 10^{5}}\) The probability of 7 successes in 12 trials with a probability of success of 0.7 is approximately 0.1585.