Question

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 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
(round to three decimal places as needed.)

Explanation:

Identify the given parameters

We are given:

• Total number of songs in the jukebox, \(N = 55\).
• Number of songs you like, \(L = 17\).
• Number of songs you do not like, \(D = 55 - 17 = 38\).
• Number of songs played sequentially without replacement, \(n = 4\).

Calculate the probability for part (a)

Using the Dependent Events and Multiplication Rule for Probability knowledge points:

$$ P(\text{like all 4}) = \frac{17}{55} \times \frac{16}{54} \times \frac{15}{53} \times \frac{14}{52} $$

$$ P(\text{like all 4}) = \frac{57,120}{8,185,320} \approx 0.006978 $$

Rounding to three decimal places:

$$ P(\text{like all 4}) \approx 0.007 $$

Calculate the probability for part (b)

Using the Dependent Events and Multiplication Rule for Probability knowledge points:

$$ P(\text{dislike all 4}) = \frac{38}{55} \times \frac{37}{54} \times \frac{36}{53} \times \frac{35}{52} $$

$$ P(\text{dislike all 4}) = \frac{1,771,560}{8,185,320} \approx 0.21643 $$

Rounding to three decimal places:

$$ P(\text{dislike all 4}) \approx 0.216 $$

Answer:

Question 1

The probability that you like the next four songs that are played is <blank>0.007</blank>.

Question 2

The probability that you do not like any of the next four songs that are played is <blank>0.216</blank>.