Take a spin An online spinner has two colored regions_blue and yellow. According to the website, the probability that the spinner lands in the blue region on any spin is $0.80$. Assume for now that this claim is correct. Suppose we spin the spinner 12 times and let $X$$=$the number of times it lands in the blue region.

a. Explain why $X$is a binomial random variable.

b. Find the probability that exactly 8 spins land in the blue region.

The total number of trials$\left(n\right)=12$

The likelihood of success $\left(p\right)=0.80$

The random variable satisfies the following requirements $X$.

1. The probability of success (p) is equal to the probability that the spinner will land in the blue region.$0.80$is fixed.

2. The number of times the spinner spins is set.

3. Spins are unrelated to one another.

4. There are two possible outcomes: either the spinner lands in the blue region or it does not.

Here, all of the binomial criteria are met.

As a result, the binomial distribution X can be said to be followed.

The total number of trials$\left(n\right)$is$12$

The likelihood of success$\left(p\right)$is$0.80$

The likelihood of 8 spins landing in the blue region can be estimated as follows:

$P(X=8)={}^{12}C_8\times (0.80{)}^8\times (1-0.80{)}^{12-8}$

$=0.1329$

As a result, the necessary probability is$0.1329$.