a radio disc jockey has 7 songs on this upcoming hours playlist: 3 are rock songs, 2 are reggae songs, and 2…
a radio disc jockey has 7 songs on this upcoming hours playlist: 3 are rock songs, 2 are reggae songs, and 2 are country songs. the disc jockey randomly chooses the first song to play, and then she randomly chooses the second song from the remaining ones. what is the probability that both songs are country songs? write your answer as a fraction in simplest form.
Answer
Explanation:
Step1: Calculate first - song probability
The probability of choosing a country song as the first song is the number of country songs divided by the total number of songs. There are 2 country songs and 7 total songs, so the probability $P_1=\frac{2}{7}$.
Step2: Calculate second - song probability
After choosing one country song as the first song, there is 1 country song left and 6 total songs left. So the probability of choosing a country song as the second song given that the first song was a country song is $P_2 = \frac{1}{6}$.
Step3: Calculate joint probability
Since these are dependent events, the probability that both songs are country songs is the product of the probabilities of each event. So $P = P_1\times P_2=\frac{2}{7}\times\frac{1}{6}=\frac{2}{42}=\frac{1}{21}$.
Answer:
$\frac{1}{21}$