[Math] Why are the probability of rolling the same number twice and the probability of rolling pairs different

diceprobability

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

  1. 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?

Best Answer

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.