Question

A pair of fair dice is tossed. Find the probability that the maximum of the two numbers is greater than 4.
$
  A.\dfrac{5}{9} \\
  B.\dfrac{4}{9} \\
  C.\dfrac{2}{9} \\
  D.{\text{None of these}} \\
 $

Answer 1

Hint: Here, in the question we need to determine the probability that the maximum of the two numbers is greater than 4 when two dice are rolled together. For this, we need to determine the total number of favorable outcomes and divide it by the total number of outcomes to get the result.

Complete Answer:
Total number of outcomes when a fair dice is rolled is 6.
Hence, the total number of outcomes when a pair of fair dice is rolled is given as:
$n(S) = 6 \times 6 = 36 - - - - (i)$

Now, the favorable number of outcome is the a number which is maximum of the two numbers is greater than 4 on either of the dices such that,
\[n(A) = \left\{ {\begin{array}{*{20}{c}}
  {\left( {5,1} \right),\left( {5,2} \right),\left( {5,3} \right),\left( {5,4} \right),\left( {5,5} \right),\left( {5,6} \right),\left( {6,1} \right),\left( {6,2} \right),\left( {6,3} \right),\left( {6,4} \right)} \\
  {\left( {6,5} \right),\left( {6,6} \right),\left( {1,5} \right),\left( {1,6} \right),\left( {2,5} \right),\left( {2,6} \right),\left( {3,5} \right),\left( {3,6} \right),\left( {4,5} \right),\left( {4,6} \right)}
\end{array}} \right\} = 20 - - - - (ii)\]
Now, as the total number of favorable outcomes is 20 and the total number of outcomes while rolling two dices is 36 then, the probability is given as:
$
  P = \dfrac{{n(A)}}{{n(S)}} \\
   = \dfrac{{20}}{{36}} \\
   = \dfrac{5}{9} $

Hence, the probability that the maximum of the two numbers is greater than 4 is $\dfrac{5}{9}$.

Hence Option A is correct.

Note: Probability means the certainty of occurring of any event. To find the probability of an experiment for which the outcomes can’t be guessed with certainty (Random experiment), two definitions are there, one is an event, and the other is sample space.
Event is the other name of the favourable outcome of any experiment while Sample space is the set of all possible outcomes of that experiment and we can say that an event will be a subset of sample space.