To determine the probability of getting exactly 3 heads out of 9 flips of a penny, we can use the binomial probability formula. The formula is given by:
[tex]\[
P(k \text{ successes}) = \binom{n}{k} p^k (1-p)^{n-k}
\][/tex]
where:
- [tex]\( n \)[/tex] is the number of trials (flips), which is 9 in this case.
- [tex]\( k \)[/tex] is the number of successes (heads), which is 3 in this case.
- [tex]\( p \)[/tex] is the probability of success (getting a head) in each trial, which is 0.5 for flipping a fair penny.
The binomial coefficient [tex]\(\binom{n}{k}\)[/tex] is calculated by:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
Plugging in the values:
- [tex]\( n = 9 \)[/tex]
- [tex]\( k = 3 \)[/tex]
- [tex]\( p = 0.5 \)[/tex]
The expression becomes:
[tex]\[
P(3 \text{ heads}) = \binom{9}{3} (0.5)^3 (1-0.5)^{9-3}
\][/tex]
Simplifying further:
[tex]\[
P(3 \text{ heads}) = \binom{9}{3} (0.5)^3 (0.5)^6
\][/tex]
Combining the powers of 0.5:
[tex]\[
P(3 \text{ heads}) = \binom{9}{3} (0.5)^{3+6}
\][/tex]
[tex]\[
P(3 \text{ heads}) = \binom{9}{3} (0.5)^9
\][/tex]
The correct expression among the given choices is:
[tex]\[
{ }_9 C_3(0.5)^3(0.5)^6
\][/tex]
This simplifies to:
[tex]\[
{ }_9 C_3(0.5)^9
\][/tex]
Hence, the correct expression that represents the probability of getting exactly 3 heads is:
[tex]\[
{ }_9 C_3(0.5)^3(0.5)^6
\][/tex]
The numerical outcome for the given problem would be [tex]\( 84.0 \)[/tex] for the combination calculation and [tex]\( 0.1640625 \)[/tex] for the probability.
