Solving for Unknown Probability in a Binomial Equation
In a binomial experiment with 10 independent trials, the probability of obtaining exactly 4 successes is 0.20. Given that X ~ Binomial(n=10, p), use the binomial probability formula \(P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}\) to set up an equation and solve for the unknown probability of success, \(p\). State your answer to 2 decimal places.
Part b
State the general term in the binomial expansion of \((1 + x)^n\) and hence identify the coefficient of \(x^k\).
Committee Selection with Minimum Girls Constraint
A club has 12 members, including 5 girls and 7 boys. In how many ways can a committee of 4 members be chosen if the committee must include at least 2 girls?