Answer
$P(all~3~bottles~are~Merlot)=\frac{1}{22}\approx0.04545$
Work Step by Step
- First bottle:
The sample space are the 12 bottles of red wine. So, $N(S_1)=12$
There are 5 Merlot. Now, consider the event "first bottle is Merlot". $N(first~bottle~is~Merlot)=5$
Using the Classical Method (page 259):
$P(first~bottle~is~Merlot)=\frac{N(first~bottle~is~Merlot)}{N(S_1)}=\frac{5}{12}$
- Second bottle:
The sample space are the 11 remaining bottles. So, $N(S_2)=11$
There are 4 remaining Merlot. Now, consider the event "second bottle is Merlot". $N(second~bottle~is~Merlot~|~first~bottle~is~Merlot)=4$
Using the Classical Method (page 259):
$P(second~bottle~is~Merlot~|~first~bottle~is~Merlot)=\frac{N(second~bottle~is~Merlot~|~first~bottle~is~Merlot)}{N(S_2)}=\frac{4}{11}$
- Third bottle:
The sample space are the 10 remaining bottles. So, $N(S_3)=10$
There are 3 remaining Merlot. Now, consider the event "third bottle is Merlot". $N(third~bottle~is~Merlot~|~first~two~bottles~are~Merlot)=3$
Using the Classical Method (page 259):
$P(third~bottle~is~Merlot~|~first~two~bottles~are~Merlot)=\frac{N(third~bottle~is~Merlot~|~first~two~bottles~are~Merlot)}{N(S_3)}=\frac{3}{10}$
Now, using the General Multiplication Rule (page 289):
$P(all~3~bottles~are~Merlot)=P(first~bottle~is~Merlot)\times P(second~bottle~is~Merlot~|~first~bottle~is~Merlot)\times P(third~bottle~is~Merlot~|~first~two~bottles~are~Merlot)=\frac{5}{12}\times\frac{4}{11}\times\frac{3}{10}=\frac{60}{1320}=\frac{1}{22}\approx0.04545$
