What are the odds of rolling a 1 and a 20 in two d20 dice?

Question

So, one of these days a friend of mine rolled two d20 dice and got a $1$ in one of them and a $20$ in the other. I was thinking, what are the odds of this happening?

Answer 1

On a D-20 dice there are 20 possible outcomes. The probability of rolling any particular face of the dice (assuming that it is a fair dice) is ${1}\over{20}$. These events are independent of each other, this means that the outcomes of the dice do not influence each other. To find a particular two dice combination we have to take the probability of the first outcome and multiply it by the probability of the second outcome. $$\frac{1}{20} * \frac{1}{20} = \frac{1}{400}$$ The chance of rolling a one and a 20 is $\frac{1}{400}$. This is actually the same probability as any particular combination of those two dice. For example, rolling a 3 and a 16 would also be a $\frac{1}{400}$ probability.

Since there are two possible ways to get the combination of one and 20, first dice is one and second is 20, or first dice is 20 and second is one. You add the two probabilities together $$\frac{1}{400} + \frac{1}{400} = \frac{2}{400} = \frac{1}{200}$$

This is the final probability of this event.

Answer 2

Imagine the two dice are different colors. One is blue and the other green.

There are $20$ possible options the blue die can roll. For each of those $20$ there are $20$ possible options there green die can roll. So there are $20*20 = 400$ possible outcomes. Each one is equally likely.

There are $2$ ways to roll a $1$ and one and a $20$ on the other. Either the blue die is $1$ and the green die is $20$, or the green die is $1$ and the blue die is $20$.

So out of the $400$ possible outcome $2$ of them are a $1$ and a $20$.

So the probability is $\frac 2{400} =\frac 1{200}$.

BTW. That is "probability". Technically "odds" is a different concept. The odds of rolling a $1$ and a $20$ are $2$ in favor to $(400 - 2)$ against. Or $2$ in favor to $398$ against. Or reduced proportionally $1$ in favor to $199$ against. Or simply, $1$ to $199$ or $1:199$.