# Different probabilities with permutation and combination?

Two fair 6-sided dice are rolled. What is the probability that both dice show the same number?

Number of ways to get the same number on both dice is 6. Total possible combinations of dice roll is $\frac{6 \cdot 6}{2} = 18$. Hence the answer is $\frac{6}{18} = \frac{1}{3}$.

But I get a different answer when I compare number of ways to get the same number on both dice to the total possible permutations of dice rolls. Total permutations of dice roll are $6 \cdot 6 = 36$. Hence the probability is $\frac{6}{36} = \frac{1}{6}$.

Why do I get different answers? When to use

$$\frac{\text{Successful permutations}}{\text{Total permutations}}$$

and when to use

$$\frac{\text{Successful combinations}}{\text{Total combinations}}$$

• Use permutations in this one, $\frac{1}{6}$ is correct Commented Feb 29, 2016 at 10:30
• To sharpen your intuition: first throw a die. Let's say it gives $5$. Then throw the second die. What is the probability that it gives a $5$ too? Commented Feb 29, 2016 at 10:43
• Oh I got it. The rolls should be considered distinct and independent to be sure about the answer. Thanks a lot for your reply :)
– AoPS
Commented Feb 29, 2016 at 11:00