Question

You look over the songs in a jukebox and determine that you like 14 of the 53 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.)

Answer 1

Answer:

a) Pr(liking four)=0.00342

b) Pr( not liking four)= 0.281

Step-by-step explanation:

The probability is mutually dependent in that the songs cannot be repeatd

total number of songs =53

number of songs likes =14

the probability of liking the first song is 14/53

since the songs cannot be repeated, the probability of  liking the next song is 13/ 52 ( because there are now only 13 songs liked on the jukebox out of 52). this continues in like manner for the third and fourth as 12/51 and 11/50 respectively .

a) the probability of liking all four will be

Pr(liking four) =Pr (liking first)* Pr(liking second)*Pr(liking third)*Pr(liking fourth)

= $\frac{14}{53} *\frac{13}{52}*\frac{12}{51} *\frac{11}{50}\\=0.00342$to 3 significant figures

Pr(liking four)=0.00342

b) total number of songs =53

number of songs likes =14

no of songs not liked is

$53-14 =39$

following the same pattern as in (a) since the events are dependent,

we have

Pr(not liking four) = Pr (not liking first)* Pr( not liking second)*Pr(not liking third)*Pr( not liking fourth)

$= \frac{39}{53} *\frac{38}{52}* \frac{37}{51}* \frac{36}{50} = 0.281$

Pr( not liking four)= 0.281 to 3 significant figures

Answer 2

Using the hypergeometric distribution, it is found that there is a:

a) 0.0034 = 0.34% probability that you like the next four songs that are​ played.

b) 0.2809 = 28.09% probability that you do not like the any of the next four songs that are​ played.

The songs are chosen without replacement, as they cannot be repeated, hence the hypergeometric distribution is used to solve this question.

What is the hypergeometric distribution formula?

The formula is:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• N is the size of the population.

• n is the size of the sample.

• k is the total number of desired outcomes.

In this problem:

• There are 53 songs, hence N = 53.

• You like 14, hence k = 14.

• 4 will be played, hence n = 4.

Item a:

The probability is P(X = 4), hence:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}$

$P(X = 4) = h(4,53,4,14) = \frac{C_{14,4}C_{39,0}}{C_{53,4}} = 0.0034$

0.0034 = 0.34% probability that you like the next four songs that are​ played.

Item b:

The probability is P(X = 0), hence:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}$

$P(X = 0) = h(0,53,4,14) = \frac{C_{14,0}C_{39,4}}{C_{53,4}} = 0.2809$

0.2809 = 28.09% probability that you do not like the any of the next four songs that are​ played.

More can be learned about the hypergeometric distribution at https://brainly.com/question/24826394