Question

a radio disc jockey has 7 songs on this upcoming hour’s 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.

Explanation:

Step1: Probability of first song (rock)

There are 2 rock songs out of 7 total. So, \( P(\text{first is rock}) = \frac{2}{7} \).

Step2: Probability of second song (country)

After choosing a rock song, 6 songs remain, with 3 country songs. So, \( P(\text{second is country}|\text{first is rock}) = \frac{3}{6} = \frac{1}{2} \).

Step3: Multiply the probabilities

Multiply the two probabilities: \( \frac{2}{7} \times \frac{1}{2} = \frac{2 \times 1}{7 \times 2} = \frac{1}{7} \)? Wait, no, wait: Wait, \( \frac{2}{7} \times \frac{3}{6} \). Let's recalculate. \( \frac{2}{7} \times \frac{3}{6} = \frac{6}{42} = \frac{1}{7} \)? Wait, no, \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \), so \( \frac{2}{7} \times \frac{1}{2} = \frac{2}{14} = \frac{1}{7} \)? Wait, no, wait: Original calculation: first, 2 rock out of 7. Then, after picking a rock, 3 country out of 6 remaining. So \( \frac{2}{7} \times \frac{3}{6} = \frac{6}{42} = \frac{1}{7} \)? Wait, no, \( 6/42 \) simplifies to \( 1/7 \)? Wait, 6 divided by 6 is 1, 42 divided by 6 is 7. Yes. Wait, but let's check again. \( \frac{2}{7} \times \frac{3}{6} = \frac{2 \times 3}{7 \times 6} = \frac{6}{42} = \frac{1}{7} \). Wait, but maybe I made a mistake. Wait, 3 country out of 6 remaining. So \( \frac{3}{6} = \frac{1}{2} \). Then \( \frac{2}{7} \times \frac{1}{2} = \frac{2}{14} = \frac{1}{7} \). Yes, that's correct.

Answer:

\(\frac{1}{7}\)