Question

section 6.2 homework score: 3/14 answered: 3/14 progress saved done question 4 0/1 pt 3 99 details using the binomial distribution with n = 10 and p = 0.4, find the following probability. round answer to four decimal places. p(x = 7)

Explanation:

Step1: Recall 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 $C(n,k)=\frac{n!}{k!(n - k)!}$.

Step2: Identify values of $n$, $k$, and $p$

Given $n = 10$, $k = 7$, and $p = 0.4$. Then $1-p=1 - 0.4 = 0.6$.

Step3: Calculate the combination $C(n,k)$

$C(10,7)=\frac{10!}{7!(10 - 7)!}=\frac{10!}{7!3!}=\frac{10\times9\times8}{3\times2\times1}=120$.

Step4: Calculate the probability

$P(X = 7)=C(10,7)\times p^{7}\times(1 - p)^{10 - 7}=120\times(0.4)^{7}\times(0.6)^{3}$.
$(0.4)^{7}=0.4\times0.4\times0.4\times0.4\times0.4\times0.4\times0.4\approx0.0016384$.
$(0.6)^{3}=0.6\times0.6\times0.6 = 0.216$.
$P(X = 7)=120\times0.0016384\times0.216\approx0.0425$.

Answer:

$0.0425$