If two identical dice are thrown simultaneously (The order of result does not matter. For example, $(2, 3)$ and $(3, 2)$ are considered same), what is the probability of getting same number on both the dice?
My attempt:
Now the reduced sample space is of size = $6+{6 \choose 2} = 6 + 15 = 21$.
Though the sample space is reduced from $36$ to $21$, the probability of getting the same number on both dice is $\frac{1}{36}$, and the probability of getting different number on both the dice is $\frac{2}{36}$.
Since we have $6$ possibilities of getting same number on both the dice, the required probability is $\frac{6}{36} = \frac{1}{6}$
Best Answer
Another way to understand that :
The probability to have the same number on the two dice is the probability that the second die gives the same number as the first die (whatever this number is). So it is equal to the probability of getting a given number with one die, so it is $$\frac{1}{6}$$