Question

An election ballot asks voters to select three city commissioners from a group of six candidates. If your aunt and father are running, what is the probability that either your aunt or your father will become a city commissioner

Answer 1

0.8 = 80% probability that either your aunt or your father will become a city commissioner.

Explanation:

The candidates are chosen from the sample without replacement, which means that the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

The probability of x successes is given by the following formula:

`P(X = x) = h(x,N,n,k) = (C_(k,x)*C_(N-k,n-x))/(C_(N,n))`

In which:

x is the number of successes.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

Combinations formula:

`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)!)`

In this question:

6 candidates, which means that
`N = 6`

2 are the aunt and father, which means that
`k = 2`

3 are chosen, which means that
`n = 3`

What is the probability that either your aunt or your father will become a city commissioner?

Probability of at least one of them being chosen, which is:

`P(X \geq 1) = 1 - P(X = 0)`

In which

`P(X = x) = h(x,N,n,k) = (C_(k,x)*C_(N-k,n-x))/(C_(N,n))`

`P(X = 0) = h(0,6,3,2) = (C_(2,0)*C_(4,3))/(C_(6,3)) = 0.2`

Then

`P(X \geq 1) = 1 - P(X = 0) = 1 - 0.2 = 0.8`

0.8 = 80% probability that either your aunt or your father will become a city commissioner.