What is the probability of rolling two dice and getting minimum value X?

Question

• Let's say I have 2 die
• I roll them

What is the probability that value on die 1 + value on die 2 is greater than 4 ?

Answer 1

Set $S=\{1,2,\cdots,6\}$ and define $A=\{(x,y)\in S^2 |\,x+y> 4\}$. We have $$\mathbb{P}(A^c)=\frac{6}{36}$$

Answer 2

There are 36 possible outcomes in total. One possibility to have a sum of 2 (1+1), 2 possibilities to have a sum of 3 (2+1, 1+2) and 3 possibilities to have a sum of 3 (1+3,3+1,2+2). Hence, six out of 36 possible outcomes are less than or equal to 4. We obtain

$$P(\text{sum} > 4) = 1 - \frac{6}{36}=\frac{30}{36}=\frac{5}{6}$$

This is all under the assumption that the dice are fair.