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

Answer:

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

Step-by-step 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.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

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