Question

If three die are thrown together, then the probability of getting five on at least one of them is:
(A) $\dfrac{{125}}{{216}}$
(B) $\dfrac{{215}}{{256}}$
(C) $\dfrac{1}{{216}}$
(D) $\dfrac{{91}}{{216}}$

Answer 1

Hint : The given question revolves around the concepts and principles of probability. One must know the basics of probability before attempting the given problem. Simplification rules and transposition are of great significance in solving such types of questions. We will first reverse the problem by finding the probability that no die shows five. Then, we will subtract the result from one as we know that the sum of probabilities of all the possibilities is always equal to $1$.

Complete step-by-step answer :
In the problem, we are given three dice and we have to find the probability that we get $5$ on at least one of the dice.
Now, we know that the sum of all the elementary probabilities of all possibilities of an event is equal to one.
So, we will find the probability of not having any five on any of the dice and then subtract it from one so as to calculate the probability of getting at least one $5$.
We know that there are six possibilities when we roll a die and we don’t want any five.
So, total number of outcomes $= 6$
Number of favorable outcomes $= 5$
So, probability that one die does not shows $5$ ${\text{ = }}\dfrac{{{\text{Number of favorable outcomes}}}}{{{\text{Total number of outcomes}}}}$
${\text{ = }}\dfrac{5}{6}$
Now, we calculate the probability of not getting $5$ on all the three dice using the multiplication rule of probability. So, we get,
Probability of not getting any five on three dice ${\text{ = }}\dfrac{5}{6} \times \dfrac{5}{6} \times \dfrac{5}{6}$
Simplifying the calculations, we get,
${\text{ = }}\dfrac{{125}}{{216}}$
Now, we can subtract this probability from one so as to get the probability of getting at least one $5$ on three dice.
So, we get the required probability as $\left( {1 - \dfrac{{125}}{{216}}} \right) = \dfrac{{91}}{{216}}$
So, option (D) is correct.
So, the correct answer is “Option D”.

Note : The sum of all possible probabilities is always one. We must know the multiplication rule of probability to solve the given questions. If the probability of occurrence of an event is a and probability of occurrence of another event is b, then the probability of occurrence of both the things is $a \times b$.