Why are the probability of rolling the same number twice and the probability of rolling pairs different?

Question

Two scenarios:
1. Using one die, roll a 6 twice.
$\frac16\times\frac16=\frac1{36}$

2. Rolling two dice roll the same number (a pair).
   $\frac6{36}=\frac16$

Why are these two probabilities different? Because the events are independent, isn't rolling a pair the same as rolling a die twice?

In a sense, rolling two dice at once is the same as rolling 1 die twice at the same time? How does this "timing" issue affect the probability?

Answer 1

In the second case, your pair can be any one of $(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)$ and you'd satisfy "obtaining a pair".

That gives you six possible pairs, each of which one has probability of occurring $\dfrac 1{36}$ gives us $$6\times \frac 1{36} = \frac 16$$

Now, if you want to know what the probability of rolling two dice simultaneously and obtaining two sixes (one prespecified pair of the six possible pairs), that would be $\dfrac 1{6\cdot 6} = \dfrac 1{36}$.

With this distinction made, yes, the probability of obtaining two sixes when rolling one die twice, and the probability of rolling two sixes simultaneously are equal.

Answer 2

Note that you are asking two different questions. In the first problem, you are asking the probability of rolling a particular number twice, while in the second problem, you are asking the probability of rolling one of the numbers twice.

The probability of rolling a $6$ twice is $$\frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$$ but so is the probability of rolling a $1$ twice, a $2$ twice, a $3$ twice, a $4$ twice, or a $5$ twice. Hence, the probability of rolling the same number twice is $$6 \cdot \frac{1}{36} = \frac{1}{6}$$