Question

A survey is being conducted in a county where 62% of the voters are Democrats and 38% are Republican. (a) What is the probability that two independently surveyed voters would both be Democrats?

Answer 1

0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats

Explanation:

For each voter, there are only two possible outcomes. Either the voter is a Democrat, or he is not. The probability of the voter being a Democrat is independent of other voters. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

62% of the voters are Democrats

This means that
`p = 0.62`

(a) What is the probability that two independently surveyed voters would both be Democrats?

This is P(X = 2) when n = 2. So

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 2) = C_(2,2).(0.62)^(2).(0.38)^(0) = 0.3844`

0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats