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. (round to three decimal places as needed.) (c) the probability that you like exactly one song is 0.556 (round to three decimal places as needed.) (d) the probability that you like both songs is 0.198. (round to three decimal places as needed.) the probability that you like neither song is 0.309 (round to three decimal places as needed.) the probability that you like exactly one song is (round to three decimal places as needed.)

Explanation:

Step1: Calculate the probability of liking exactly one song when replacement is allowed

The probability of liking a song is $p=\frac{4}{9}\approx0.444$ and the probability of not - liking a song is $q = 1-\frac{4}{9}=\frac{5}{9}\approx0.556$.
There are two cases: like the first and not like the second or not like the first and like the second.

Step2: Calculate the probability for each case

For the first case (like first, not like second), the probability is $P_1=\frac{4}{9}\times\frac{5}{9}$.
For the second case (not like first, like second), the probability is $P_2=\frac{5}{9}\times\frac{4}{9}$.

Step3: Calculate the total probability

The probability of liking exactly one song is $P = P_1 + P_2=2\times\frac{4}{9}\times\frac{5}{9}=\frac{40}{81}\approx0.494$.

Answer:

0.494