Probability of rolling a pair of dice

Question

Suppose we have two fair dice and rolled them.

Let

• $A$ be the event "the sum of the two dice is equal to $3$";
• $B$ be the event "the sum of the two dice is equal to $7$";
• $C$ be the event "at least one dice shows $1$".

How to calculate $P(A \mid C)$?

In this case can we say that $A$ and $C$ are independent? Can we say that $B$ and $C$ are independent?

Answer 1

$P(C)$ is actually $\frac{11}{36}$ – $11$ of the $36$ possible rolls show at least one 1 (don't forget to consider the double-1 case!). $P(A\cap C)=\frac2{36}$ since only 1-2 and 2-1 have at least one 1 and sum to $3$. Thus $$P(A|C)=\frac{P(A\cap C)}{P(C)}=\frac{2/36}{11/36}=\frac2{11}$$

Answer 2

If one of the dices shows $1$ then there are only two ways to get $3.$ If both the dices show $1$ then there is no chance of getting $3.$ So the required probability is $\frac {2} {11}.$ Because when $C$ has already occurred then the reduced sample space has $11$ elements where at least one of the events is $1$ which are $$\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(3,1),(4,1),(5,1),(6,1) \}$$ and only two of them suit your purpose which are $$\{(1,2),(2,1) \}.$$