Question

suppose you just purchased a digital music player and have put 9 tracks on it. after listening to them you decide that you like 4 of the songs. with the random - feature on your player, each of the 9 songs is played once in random order. find the probability that among the first two songs played (a) you like both of them. would this be unusual? (b) you like neither of them. (c) you like exactly one of them. (d) redo (a)-(c) if a song can be replayed before all 9 songs are played.
(a) the probability that you like both songs is 0.167 (round to three decimal places as needed.) would it be unusual for you to like both of the songs? no yes
(b) the probability that you like neither song is (round to three decimal places as needed.)

Explanation:

Step1: Calculate probability of not - liking first song

There are 9 songs in total and 4 songs you like, so 9 - 4=5 songs you don't like. The probability of not liking the first song is $\frac{5}{9}$.

Step2: Calculate probability of not - liking second song given not - liking first

Since the songs are played without replacement, after not liking the first song, there are 8 songs left and 4 songs you like, so 8 - 4 = 4 songs you don't like. The probability of not liking the second song given that you didn't like the first is $\frac{4}{8}$.

Step3: Calculate probability of not liking either song

By the multiplication rule for independent events (in the context of sequential non - replacement selection), the probability that you like neither song is $\frac{5}{9}\times\frac{4}{8}=\frac{20}{72}\approx0.278$.

Answer:

0.278