Question

Choose the correct option from the below given options by solving the following question:
Two dice are thrown together. The probability of getting the same number on both dice is:
A. $\dfrac{1}{2}$
B. $\dfrac{1}{3}$
C. $\dfrac{1}{6}$
D. $\dfrac{1}{{12}}$

Answer 1

We have to find the probability of getting the same number on both the dice when rolled together. For that, we need to write the sample space and then pick the pairs which have the same number. Then, we find the probability by using the formula.

Complete step-by-step solution:
Given,
Two dice are thrown together. We have to find the probability of getting the same number on both the dice.
Now, each dice contains six faces. If two dice are thrown together, we can calculate the outcome by squaring the single dice outcome.
Number of all the possible total outcomes $= {6^2}$
$\Rightarrow$ Number of all the possible total outcomes $= 36$
Let us write all the outcomes possible to fish out the outcomes where both the dice show the same number.
Total sample space$= 36$. They are,

\left( {1,1} \right),\left( {1,2} \right),\left( {1,3} \right),\left( {1,4} \right),\left( {1,5} \right),\left( {1,6} \right) \\\ \left( {2,1} \right),\left( {2,2} \right),\left( {2,3} \right),\left( {2,4} \right),\left( {2,5} \right),\left( {2,6} \right) \\\ \left( {3,1} \right),\left( {3,2} \right),\left( {3,3} \right),\left( {3,4} \right),\left( {3,5} \right),\left( {3,6} \right) \\\ \left( {4,1} \right),\left( {4,2} \right),\left( {4,3} \right),\left( {4,4} \right),\left( {4,5} \right),\left( {4,6} \right) \\\ \left( {5,1} \right),\left( {5,2} \right),\left( {5,3} \right),\left( {5,4} \right),\left( {5,5} \right),\left( {5,6} \right) \\\ \left( {6,1} \right),\left( {6,2} \right),\left( {6,3} \right),\left( {6,4} \right),\left( {6,5} \right),\left( {6,6} \right) \\\ \end{gathered} $$ In the above given pairs, the pairs which have the same number on both dice are; $$\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right),\left( {4,4} \right),\left( {5,5} \right),\left( {6,6} \right)$$ Therefore, the number of outcomes for the required condition$$ = 6$$ The probability that there will be the same number on two dice$$ = \dfrac{6}{{36}}$$ $$ \Rightarrow $$ The probability that there will be the same number on two dice $$ = \dfrac{1}{6}$$ Therefore, the required probability $$ = \dfrac{1}{6}$$. **The correct option is C.** **Note:** Probability is also a branch of mathematics which tells us how likely an event is to occur or it also tells us how much percentage or how likely a given proposition is true. The probability only ranges between $$0$$ and $$1$$. If the sample space is $$S$$ and there is an event $$A$$ occurring which has $$n$$elements, we can say that the probability of $$A$$ is given as: $$P\left( A \right) = \dfrac{S}{n}$$.