Answer
$P(I~like~the~two~first~songs)=P(I~like~the~first~song)\times P(I~like~the~second~song)=\frac{5}{13}\times\frac{5}{13}=\frac{25}{169}\approx0.1479$
$P(I~do~not~like~the~two~first~songs)=P(I~do~not~like~the~first~song)\times P(I~do~not~like~the~second~song)=\frac{8}{13}\times\frac{8}{13}=\frac{64}{169}\approx0.3787$
$P(I~like~exactly~one~of~the~two~first~songs)=\frac{80}{169}\approx0.4734$
Work Step by Step
(a) - First music:
The sample space are the 13 tracks. So, $N(S_1)=13$
I like 5 songs. Now, consider the event "I like the first song". $N(I~like~the~first~song)=5$
Using the Classical Method (page 259):
$P(I~like~the~first~song)=\frac{N(I~like~the~first~song)}{N(S_1)}=\frac{5}{13}$
The events "I like the first song" and "I like the second song" are independent events.
- Second music:
The sample space are the 13 tracks. So, $N(S_2)=13$
There are 5 songs that I like. Now, consider the event "I like the second song". $N(I~like~the~second~song)=5$
Using the Classical Method (page 259):
$P(I~like~the~second~song)=\frac{N(I~like~the~second~song)}{N(S_2)}=\frac{5}{13}$
Now, using the Multiplication Rule for Independent Events (page 282):
$P(I~like~the~two~first~songs)=P(I~like~the~first~song)\times P(I~like~the~second~song)=\frac{5}{13}\times\frac{5}{13}=\frac{25}{169}\approx0.1479$
(b) - First music:
The sample space are the 13 tracks. So, $N(S_1)=13$
I do not like $13-5=8$ songs. Now, consider the event "I do not like the first song". $N(I~do~not~like~the~first~song)=8$
Using the Classical Method (page 259):
$P(I~do~not~like~the~first~song)=\frac{N(I~do~not~like~the~first~song)}{N(S_1)}=\frac{8}{13}$
The events "I do not like the first song" and "I do not like the second song" are independent events.
- Second music:
The sample space are the 13 tracks. So, $N(S_2)=13$
There are $13-5=8$ songs that I do not like. Now, consider the event "I do not like the second song". $N(I~do~not~like~the~second~song)=8$
Using the Classical Method (page 259):
$P(I~do~not~like~the~second~song)=\frac{N(I~do~not~like~the~second~song)}{N(S_2)}=\frac{8}{13}$
Now, using the Multiplication Rule for Independent Events (page 282):
$P(I~do~not~like~the~two~first~songs)=P(I~do~not~like~the~first~song)\times P(I~do~not~like~the~second~song)=\frac{8}{13}\times\frac{8}{13}=\frac{64}{169}\approx0.3787$
(c) The events "I like the two first songs" and "I do not like the two first songs" are mutually exclusives (disjoint events).
Using the Addition Rule for Disjoint Events (page 270):
$P(I~like~the~two~first~songs~or~I~do~not~like~the~two~first~songs)=P(I~like~the~two~first~songs)+P(I~do~not~like~the~two~first~songs)=\frac{25}{169}+\frac{64}{169}=\frac{89}{169}$
The event "I like exactly one of the two first songs" is the complement of "I like the two first songs or I do not like the two first songs".
Using the Complement Rule (see page 275):
$P(I~like~exactly~one~of~the~two~first~songs)=1-P(I~like~the~two~first~songs~or~I~do~not~like~the~two~first~songs)=1-\frac{89}{169}=\frac{80}{169}\approx0.4734$
