Question

Joey has a playlist that he listens to on his walk to school each morning. The songs are played at random, and none of the songs are repeated until they've all played. His playlist contains 5 country songs, 3 alternative songs, and 4 hip-hop songs. What is the probability that the first two songs that play are both country songs?

Answer 1

0.1515 = 15.15% probability that the first two songs that play are both country songs

Explanation:

Songs are not repeated, which means that they are chosen without replacement, and the hypergeometric distribution is used.

Hypergeometric distribution:

The probability of x sucesses 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 sucesses.

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)!)`

His playlist contains 5 country songs, 3 alternative songs, and 4 hip-hop songs. What is the probability that the first two songs that play are both country songs?

5 + 3 + 4 = 12 total musics, which means that
`N = 12`

5 country, which means that
`k = 5`

First two, which means that
`n = 2`

The probability is P(X = 2). So

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

`P(X = 2) = h(2,12,2,5) = (C_(5,2)*C_(7,0))/(C_(12,2)) = 0.1515`

0.1515 = 15.15% probability that the first two songs that play are both country songs