Question

aron flips a penny 9 times. which expression represents the probability of getting exactly 3 heads?
$p(k\text{ successes}) = _nc_kp^k(1 - p)^{n - k}$
$_nc_k=\frac{n!}{(n - k)!k!}$
$\circ _9c_3(0.5)^3(0.5)^6$
$\circ _9c_3(0.5)^3$
$\circ _9c_3(0.5)^3(0.5)^9$
$\circ _9c_6(0.5)^6$

Explanation:

Step1: Identify values for formula

The binomial - probability formula is \(P(k\text{ successes})={}_{n}C_{k}p^{k}(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, and \(p\) is the probability of success on a single trial. Here, \(n = 9\) (number of coin - flips), \(k = 3\) (number of heads), and for a fair coin \(p=0.5\) (probability of getting a head on a single flip), and \(1 - p = 0.5\).

Step2: Substitute values into formula

Substitute \(n = 9\), \(k = 3\), and \(p = 0.5\) into the formula. We get \(P(3\text{ heads})={}_{9}C_{3}(0.5)^{3}(1 - 0.5)^{9 - 3}={}_{9}C_{3}(0.5)^{3}(0.5)^{6}\).

Answer:

\({}_{9}C_{3}(0.5)^{3}(0.5)^{6}\) (corresponding to the first option in the multiple - choice list)