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 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 it 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.

Answer 1

ANSWER TO PART (a) you like both of them

we have a total of 9 tracks among which we like only 4 of the tracks. Now we are playing the first 2 songs.

here, we have two events

event A= we like the first song

event B=we like the second song too

now, we need to find the probability of events Aand B occuring together,

P(E and F) = P(E)*P(E/F)

By general multiplication rule P(E and F) is the product of probability of occurence of event A and probability of occurence of event B given that A has already occured.

out of 9 tracks we like only 4

hence, P(A) = 4/9

now since A has already occured we are left with a total of 8 tracks among which we like only 3, since 1 song that we like has already been played as 1st song

hence, P(B) = 3/8

therefore, P(A and B) = 4/9 * 3/8 = 1/6 = 0.167

now, would it be unusual?

recall that, the probability of an event is said to be unusual only if the probability of the said event is less than 5% or is less than 0.05.

here the probability is 0.167 which is definitely greater than 0.05, hence, we can say that it would not be unsual.

ANSWER TO PART(b) you like neither of them

we have a total of 9 tracks among which we like only 4 of the tracks. Now we are playing the first 2 songs.

here, we have two events

event A= we do not like the first song

event B= we do not like the second song either

now, we need to find the probability of events A and B occuring together,

P(A and B) = P(A)*P(A/B)

By general multiplication rule P(A and B) is the product of probability of occurence of event A and probability of occurence of event B given that A has already occured.

out of 9 tracks we do not like 5 tracks

hence, P(A) = 5/9

now since A has already occured we are left with a total of 8 tracks among which we do not like only 4, since 1 song that we do not like has already been played as 1st song

hence, P(B) = 4/8

therefore, P(A and B) = 4/9 * 4/8 = 1/6 = 0.22

ANSWER TO PART(c) You like exactly one of them

While playing the first two songs, we have the following possibilties:

1. we like both of the songs

2. we do not like either of then

3. we like exactly one song among the two songs played

hence,the probabilities of above three possibilities will add upto 1.

since, we have already computed the probabilities of first two possibilities, we can easily compute the probability of the third one,

P(like both songs) + P(don not like both the songs) + P(like exactly one of them) = 1

0.16 + 0.22 + P(ike exactly one of them) = 1

P(ike exactly one of them) = 1-(0.16+0.22)

P(ike exactly one of them) = 1-0.38

P(ike exactly one of them) = 0.62

ANSWER TO PART(d) songs can be replayed

we have a total of 9 tracks among which we like only 4 of the tracks. Now we are playing the first 2 songs.

replaying basically means that the events are independent of each other,

i.e., P(A and B) = P(A)*P(B)

Therefore, (a) you like both of them. would it be unusual?

event A= we do not like the first song

event B= we do not like the second song either

P(A) = 4/9

now, since replaying is allowed you are still left with 9 tracks among which 4 you like

hence, P(B) = 4/9

therefore, P(A and B) = 4/9 * 4/9 = 0.197

now, would it be unusual?

recall that, the probability of an event is said to be unusual only if the probability of the said event is less than 5% or is less than 0.05.

here the probability is 0.197 which is definitely greater than 0.05, hence, we can say that it would not be unsual.

(b) you like neither of them?

we have a total of 9 tracks among which we like only 4 of the tracks. Now we are playing the first 2 songs.

here, we have two events

event A= we do not like the first song

event B= we do not like the second song either

P(A) = 5/9

now, since replaying is allowed you are still left with 9 tracks among which 5 you do not like

hence, P(B) = 5/9

therefore, P(A and B) = 5/9 *5/9 = 25/81 = 0.308

P(A and B) = 0.31(rounded off)

(c) you like exactly one of them?

While playing the first two songs, we have the following possibilties:

1. we like both of the songs

2. we do not like either of then

3. we like exactly one song among the two songs played

hence,the probabilities of above three possibilities will add upto 1.

since, we have already computed the probabilities of first two possibilities, we can easily compute the probability of the third one,

P(like both songs) + P(don not like both the songs) + P(like exactly one of them) = 1

0.197 + 0.31 + P(ike exactly one of them) = 1

P(ike exactly one of them) = 1-(0.197+0.31)

P(ike exactly one of them) = 1-0.507

P(ike exactly one of them) = 0.51 (rounded off)