Question

If you roll two fair six-sided dice what's the probability that both dice show an odd number?

Answer 1

Finding the probability of a given experiment:

When a dice is thrown, the possible outcomes are: $\left\{1,2,3,4,5,6\right\}$

Similarly, when two dice are thrown, there will be $6\times 6=36$ outcomes.

$S=\left\{\left(1,1\right),\left(1,2\right),\left(1,3\right)...(6,6)\right\}$.

Among these, outcomes with both dice showing odd numbers are $9$, which are: $A=\left\{\left(1,1\right),\left(1,3\right),\left(1,5\right),\left(3,1\right),\left(3,3\right),(3,5),\left(5,1\right),\left(5,3\right),(5,5)\right\}$

Now, The probability of an experiment is:

$P\left(A\right)=\frac{\mathrm{Number}\mathrm{of}\mathrm{favourable}\mathrm{outcomes}}{\mathrm{Number}\mathrm{of}\mathrm{total}\mathrm{outcomes}}$

That is:

$\begin{array}{rcl}P\left(A\right)& =& \frac{9}{36}\\ & =& \frac{1}{4}\end{array}$

Therefore, when two fair six-sided dice are thrown, the probability that both dice show an odd number is $\frac{\mathbf{1}}{\mathbf{4}}$.