Dice Rolling 4d10 with a twist

Question

Suppose I roll two 10-sided dice, 1 die has numbers o, 10, 20, 30 etc to 90. The second die has numbers 0, 1 ,2 etc to 9. These dice are used to create a number from 1 to 100 - example: the first die rolls a 40, the second die rolls a 9 = the number 49. The number 100 is garnered by both dice rolling 0.

If I do this twice, i.e. two people rolling at the same time (realizing that "at the same time" doesn't really apply - dice could care less what time it is)

What is the probability and/or odds of both set of dice coming out the same number? (without deciding what that number is before the roll)

AND

What is the probability that both sets of dice would match a specific number say, 49?

AND

Are the probabilities any different if you were to use a single 100 faced die?

Please feel free to explain in detail, I am not a math novice, but am not well versed in statistics.

Answer 1

We have not been told about the physical properties of these dice. We will assume, without justification, that the dice are "fair" (all $10$ sides are equally likely).

The numbers $1$ to (weird) $100$ are equally likely. Whatever Alicia picks, the probability Beti's dice roll matches it is $\frac{1}{100}$.

As for both getting (say) $49$, the probability is $\frac{1}{100}\cdot\frac{1}{100}$.

Since with the two dice all numbers are equally likely, they are the equivalent of a single "fair" $100$-sided die.

Answer 2

To clear out the second question first, since it makes the first easier: the probabilities are no different than if you'd rolled a single d100 (note that this is because the two d10 that make up a single roll are distinct - before you even start you know which is the 'tens' and which is the 'ones'). The easiest way to see this is to lay out the outcomes of the two dice as a $10\times10$ grid and put the resulting number into the grid there; this makes it easy to see that every possible way of rolling two dice corresponds uniquely to a single roll of a d100.

With that said, the probability for the two dice to match (as André has noted in his answer) is $\frac{1}{100}$; this can be thought of as taking the first person's roll as a given and figuring out the odds that the second person's roll matches it, but you can also calculate it directly; there are $100\times100=10000$ possible outcomes for the two rolls, of which exactly $100$ match, giving a probability of $\frac{100}{10000}=\frac{1}{100}$.