You look over the songs in a jukebox and determine that you like 17 of the 55 songs.
(a) What is the probability that you like the next four songs that are played? (Assume a song cannot be repeated.)
(b) What is the probability that you do not like the any of the next four songs that are played? (Assume a song cannot be repeated.)
(a) The probability that you like the next four songs that are played is 0.007 .
(Round to three decimal places as needed.)
(b) The probability that you do not like any of the next faur songs that are played is (Round to three decimal places as needed.)
Answer
Final Answer: (a) The probability that you like the next four songs that are played is \(\boxed{0.007}\) (rounded to three decimal places). (b) The probability that you do not like any of the next four songs that are played is \(\boxed{0.216}\) (rounded to three decimal places).
Step 1 ：The total number of songs in the jukebox is 55, and you like 17 of them.
Step 2 ：For the first question, we need to calculate the probability of liking the next four songs. Since the songs cannot be repeated, this is a case of probability without replacement.
Step 3 ：The probability of liking the first song is \(\frac{17}{55}\). After one song is played, the total number of songs becomes 54, and the number of liked songs becomes 16. So, the probability of liking the second song is \(\frac{16}{54}\). Similarly, the probabilities of liking the third and fourth songs are \(\frac{15}{53}\) and \(\frac{14}{52}\) respectively.
Step 4 ：The total probability of liking the next four songs is the product of these four probabilities, which is approximately \(0.007\).
Step 5 ：For the second question, we need to calculate the probability of not liking any of the next four songs. The total number of songs is 55, and the number of disliked songs is 55 - 17 = 38.
Step 6 ：The probability of not liking the first song is \(\frac{38}{55}\). After one song is played, the total number of songs becomes 54, and the number of disliked songs becomes 37. So, the probability of not liking the second song is \(\frac{37}{54}\). Similarly, the probabilities of not liking the third and fourth songs are \(\frac{36}{53}\) and \(\frac{35}{52}\) respectively.
Step 7 ：The total probability of not liking the next four songs is the product of these four probabilities, which is approximately \(0.216\).
Step 8 ：Final Answer: (a) The probability that you like the next four songs that are played is \(\boxed{0.007}\) (rounded to three decimal places). (b) The probability that you do not like any of the next four songs that are played is \(\boxed{0.216}\) (rounded to three decimal places).
