In the United States, voters who are neither Democrat nor Republican are called Independents. It is believed that 10% of all voters are Independents A survey asked 25 people to identify themselves as Democrat, Republican, or Independent.
a. What is the probability that none of the people are Independent?b. What is the probability that fewer than five people are Independent?c. What is the probability that more than two people are Independent?

Since [tex]n=25[/tex] is relatively small, and there are only two outcomes tested(being independent or not) the random variable [tex]X[/tex] can be assumed to follow a binomial distribution with [tex]p=0.10[/tex]

a. Binomial distribution is defined as: [tex]P(X=x)={n\choose x}{p)^x(1-p)^n^-^x\\[/tex]

[tex]\therefore P(X=0)={25\choose 0}(0.1^0)(1-0.1)^2^5\\=0.07179[/tex]

b.Probability of fewer than 5people are independent:

[tex]P(X<5)=P(X=4)+P(X=3)+P(X=2)+P(X=1)+P(X=0)\\\ \ ={25\choose 4}(0.1^4)(0.9^2^1)+....+{25\choose 0}(0.1^0)(0.9^2^5)\\=0.13842+0.22650+0.26589+0.19942+0.07179\\=0.90202[/tex]

c.probability than more than two people are independent

[tex]P(X>2)=1-P(X=0)-P(X=1)-P(X=2)\\=1-0.07179-0.19942-0.26589\\=0.4629[/tex]