Question

A fair $6$ - sided die is rolled three times.

Find the probability that the sum of the three dice is $4$?

Answer 1

Find the probability for the sum of the three dice is $\mathbf{4}$:

Given, a fair $6$ - sided die rolled three times.

Step- 1: Finding the possible outcomes.

$\begin{array}{rcl}\mathrm{Total}\mathrm{possible}\mathrm{outcomes}\mathrm{are}& =& 6^3\\ & =& 216\end{array}$

Now, get at least $1$ on each die, it adds up to a sum of $3$.

In order to get a sum of $4$ on three rolls of a die the required set of outcomes is $1,1,2$.

Step 2. Finding favorable outcomes.

$\mathrm{Favourable}\mathrm{outcomes}\mathrm{are}\left\{1,1,2\right\},\left\{1,2,1\right\},\left\{2,1,1\right\}$.

Number of favorable outcomes are $3$

Step 3. Finding the probability of getting the sum of three dice as $4$:

Let, the probability be $\alpha$.

The formula for probability is represented by,

$\mathrm{\alpha }=\frac{\mathrm{numbers}\mathrm{of}\mathrm{favourable}\mathrm{outcome}}{\mathrm{numbers}\mathrm{of}\mathrm{total}\mathrm{outcome}}$

Now, putting numbers of favorable outcomes as $3$ and numbers of total outcomes as $216$

$\begin{array}{rcl}\alpha & =& \frac{3}{216}\\ & =& \frac{1}{72}\end{array}$.

Hence, the probability for the sum of the three dice is $\mathbf{4}$ is $\frac{\mathbf{1}}{\mathbf{72}}$.