a radio disc jockey has 7 songs on this upcoming hours playlist: 2 are rock songs, 2 are reggae songs, and 3…

a radio disc jockey has 7 songs on this upcoming hours playlist: 2 are rock songs, 2 are reggae songs, and 3 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 the first song is a rock song and the second is a country song? write your answer as a fraction in simplest form.

a radio disc jockey has 7 songs on this upcoming hours playlist: 2 are rock songs, 2 are reggae songs, and 3 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 the first song is a rock song and the second is a country song? write your answer as a fraction in simplest form.

Answer

Explanation:

Step1: Calculate probability of first - rock song

The probability of choosing a rock song first is the number of rock songs divided by the total number of songs. There are 2 rock songs and 7 total songs, so the probability $P(R_1)=\frac{2}{7}$.

Step2: Calculate probability of second - country song

After choosing a rock song first, there are 6 remaining songs. There are 3 country songs. So the probability of choosing a country song second given that a rock song was chosen first is $P(C_2|R_1)=\frac{3}{6}=\frac{1}{2}$.

Step3: Calculate joint probability

By the multiplication rule for conditional probability $P(R_1\cap C_2)=P(R_1)\times P(C_2|R_1)$. Substitute the values: $\frac{2}{7}\times\frac{1}{2}=\frac{2\times1}{7\times2}=\frac{1}{7}$.

Answer:

$\frac{1}{7}$