Consider two dice, each one having one face showing the letter a, two faces showing the letter b, and the remaining three faces showing the letter c. You roll each >die once, independently of the other die.
What is the sample space?
Define the events:
A = "at least one of the two dice shows the letter b on its top face"
B = "both dice show the same letter on their top faces".
Determine Pr(A), Pr(B), and Pr(A | B).
S={(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,a),(c,c)}
S={(i,j): a≤i≤c a≤j≤c}
∴|S|=9
A={(a,b),(b,a),(b,b),(b,c),(c,b)}
∴|A|=5
B={(a,a),(b,b),(c,c)}
∴|B|=3
∴Pr(A)=5/9
∴Pr(B)=3/9
Pr(A∩B)= 5/9 x 3/9 = 5/27
∴Pr(A│B)=5/9
Am I correct?
Hint:
You should take into consideration that the outcomes in your sample space are not equiprobable since the individual die rolls of a, b, c are not equiprobable. For instance the probability of (c,c) is $\frac9{36}$ while the probability of (a,a) is just $\frac{1}{36}$