Question

a fair coin is flipped seven times. what is the probability of the coin landing heads up at most five times?
a. 94%
b. 19%
c. 96%
d. 98%

Explanation:

Step1: Use binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 7$, $p=\frac{1}{2}$, and $1-p=\frac{1}{2}$.

Step2: Calculate the probability of complementary events

The probability of getting at most 5 heads is $P(X\leq5)=1 - P(X = 6)-P(X = 7)$.
For $k = 6$:
$C(7,6)=\frac{7!}{6!(7 - 6)!}=\frac{7!}{6!1!}=7$
$P(X = 6)=C(7,6)\times(\frac{1}{2})^{6}\times(\frac{1}{2})^{7 - 6}=7\times(\frac{1}{2})^{7}=\frac{7}{128}$
For $k = 7$:
$C(7,7)=\frac{7!}{7!(7 - 7)!}=1$
$P(X = 7)=C(7,7)\times(\frac{1}{2})^{7}\times(\frac{1}{2})^{7 - 7}=(\frac{1}{2})^{7}=\frac{1}{128}$

Step3: Calculate $P(X\leq5)$

$P(X\leq5)=1-\frac{7}{128}-\frac{1}{128}=1-\frac{7 + 1}{128}=1-\frac{8}{128}=1-\frac{1}{16}=\frac{15}{16}=0.9375\approx94\%$

Answer:

A. 94%