a number cube with six sides is rolled seven times. the probability of getting a 4 exactly twice is

Explanation:

Step1: Identify 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 successful trials, $p$ is the probability of success in a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$.

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

When rolling a number - cube, $n = 7$ (the number of rolls), $k = 2$ (the number of times we want to get a 4), and $p=\frac{1}{6}$ (the probability of getting a 4 in a single roll), $1 - p=\frac{5}{6}$.

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

$C(7,2)=\frac{7!}{2!(7 - 2)!}=\frac{7!}{2!5!}=\frac{7\times6}{2\times1}=21$.

Step4: Calculate the probability

$P(X = 2)=C(7,2)\times(\frac{1}{6})^{2}\times(\frac{5}{6})^{7 - 2}=21\times\frac{1}{36}\times(\frac{5}{6})^{5}=21\times\frac{1}{36}\times\frac{3125}{7776}=\frac{21\times3125}{36\times7776}=\frac{65625}{279936}\approx0.234$.

Answer:

$\frac{65625}{279936}\approx0.234$