Question

If I have a $20$ sided dice and roll it twice, what are the odds that I will roll the same number twice?

Answer 1

Find the odds that the same number rolls twice, if $\mathbf{20}$ sided dice rolls twice.

Step 1: Find the total number of outcomes.

Given that, $20$ sided dice and roll it twice
When the dice roll has $\text{'}\mathrm{n}\text{'}$ faced twice, then the total number of outcomes will be $(\mathrm{n}\times \mathrm{n})$.
So, the total number of the outcome's for a $20$ sided dice:

$\begin{array}{rcl}\mathrm{n}\times \mathrm{n}& =& 20\times 20\\ & =& 400\end{array}$

$$\begin{gathered} (1,1),(1,2),(1,3),(1,4),(1,5).........(1,20) \\ (2,1),(2,2),(2,3),(2,4),(2,5).........(2,20) \\ (3,1),(3,2),(3,3),(3,4),(3,5).........(3,20) \\ (4,1),(4,2),(4,3),(4,4),(4,5).........(4,20) \\ (5,1),(5,2),(5,3),(5,4),(5,5).........(5,20) \\ . \\ . \\ (20,1),(20,2),(20,3),(20,4),(20,5),(20,6).......(20,20) \end{gathered}$$

So, there are $400$ pairs of total outcomes.

As it can be seen the pairs which have the same numbers are:

$(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8),(9,9),(10,10),(11,11),(12,12),(13,13),(14,14),(15,15),(16,16),(17,17),(18,18),(19,19),(20,20)$

So, the number of successful outcomes are $20$.

Step 2: Find the odds that the same number rolls twice.

Use the probability formula:

$\mathrm{P}\left(\mathrm{A}\right)=\frac{\mathrm{Number}\mathrm{of}\mathrm{successful}\mathrm{outcomes}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{possible}\mathrm{outcomes}}$

Substitute the known values in the above formula:

$\begin{array}{rcl}\mathrm{P}\left(\mathrm{A}\right)& =& \frac{20}{400}\\ & =& \frac{2}{40}\\ & =& \frac{1}{20}\end{array}$

Hence, the odds of getting the same number twice if $\mathbf{20}$ sided dice is rolled twice is$\frac{\mathbf{1}}{\mathbf{20}}.$