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The binomial distribution determines the probability of looking at a specific quantity of a hit results in a specific quantity of trials.

We are given,

$8\%$of students at a local high school participate in after-school sports all four years of high school and a group of $60$seniors is randomly chosen.

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case the random variable X is the number who participated in after-school sports all four years of high school.

Make the list of values that you want to use X may take on.

As we can see there is an upper bound for the situation at hand, $60$, then X is given by:

$X=0,1,2,.........,60.$

The random variable is distributed by the data provided X will keep the track of online offerings.

According to the given information, $8\%$of students at a local high school participate in after-school sports all four years of high school and a group of $60$seniors is randomly chosen.

The probability distribution of binomial distribution has two parameters role="math" localid="1649086363932" $$\begin{gathered} n=60 \\ numberoftrials \\ p=0.08 \\ probabilityofsuccess \end{gathered}$$

The binomial distribution is of the form: $X~B(n,p)$

Therefore,

$X~B(60,0.08)$

The expected binomial distribution is calculated as:

$\mathrm{μ}=np$

$\mathrm{μ}$is the number of seniors are expected to have participated in after-school sports all four years of high school,

$n=60$

$p=0.08$

Therefore, the number of seniors are expected to have participated in after-school sports all four years of high school is:

$$\begin{gathered} \mathrm{μ}=np \\ \mathrm{μ}=60×0.08 \\ \mathrm{μ}=4.8 \end{gathered}$$

As we have found the number of seniors that have participated in after-school sports all four years of high school is $4.8$

Therefore, the probability of the seniors who have not participated at all in after-school sports all four years of high school is $0.$

$P\left(0\right)=0.0067$

Using the $TI-83$calculator the probability of four or that five of the seniors participated in after-school sports all four years of high school is

$P\left(4\right)=0.1873$and $P\left(5\right)=0.1824$

We can conclude that,