# Probability of rolling a pair of dice

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?

• I don't see how $A \cap C$ and $B \cap C$ can be independent, since they're mutually exclusive and each has non-zero probability. Did you mean something else? – Brian Tung Mar 13 '19 at 7:18
• Dice is already plural. The singular is die. So it's one die, 2 dice, 3 dice... – Davor Mar 13 '19 at 11:01

$$P(C)$$ 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 $$P(A|C)=\frac{P(A\cap C)}{P(C)}=\frac{2/36}{11/36}=\frac2{11}$$
• Thank you. Can I say that $P(B|C)=\frac{2}{11}$? – Ali J. Mar 13 '19 at 6:44
• @AliJ Yes, since $P(B\cap C)=P(A\cap C)$. – Parcly Taxel Mar 13 '19 at 6:48
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 $$\{(1,2),(2,1) \}.$$
• You mean that $P(A|C)=\frac{1}{18}$? – Ali J. Mar 13 '19 at 6:16
• May you please help about $P(B|C)$? – Ali J. Mar 13 '19 at 6:16
• Same thing happens for $P(B \mid C).$ – Dbchatto67 Mar 13 '19 at 6:18
• Can't we use the rule $P(A|C)=\frac{P(A.C)}{P(A)}$ – Ali J. Mar 13 '19 at 6:19
• Try to think it on your own. If the event $C$ has already occured then in how many ways can the event $B$ occur? – Dbchatto67 Mar 13 '19 at 6:19