Question

which expression represents the probability of rolling a 5 exactly three times in ten rolls of a number cube with six sides? 10c3(1/6)^3(1/6)^7 10c3(1/2)^3(1/2)^7 10c3(1/6)^3(5/6)^7 10c3(1/6)^7(5/6)^3

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 $p$

Here, $n = 10$ (number of rolls), $k = 3$ (number of times rolling a 5), and since the die has 6 sides, the probability of rolling a 5 in a single roll $p=\frac{1}{6}$, and $1 - p = 1-\frac{1}{6}=\frac{5}{6}$.

Step3: Calculate the binomial coefficient

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

Step4: Substitute values into the formula

$P(X = 3)=C(10,3)\times(\frac{1}{6})^{3}\times(\frac{5}{6})^{10 - 3}=_{10}C_{3}(\frac{1}{6})^{3}(\frac{5}{6})^{7}$.

Answer:

$_{10}C_{3}(\frac{1}{6})^{3}(\frac{5}{6})^{7}$ (assuming the correct option among the ones in the image is the one with this form)