Question

Four people are playing a game in which each person rolls a six-sided die at the same time. Find the probability that at least two of the people roll the same number.

Answer 1

Hint:
Here, we will use the concept of probability to solve this question. The probability of an event is the chance that the event occurs. We will also use the concept of probability of complementary events. In this question, we will calculate the probability that no two people roll the same number and subtract it from 1 to get the answer.

Complete step by step solution:
First, we will find the total number of possible outcomes and the number of favourable outcomes.
The possible outcomes when a die is rolled are 1, 2, 3, 4, 5, or 6.
Thus, we observe that there are 6 possible outcomes.
Each of the four people can get any of the 6 possible outcomes.
Therefore, we can get the total number of ways as
\[6 \times 6 \times 6 \times 6 = 1296\]
Now, we assume that no two people roll the same number.
This means that the first person can roll any of the 6 numbers.
Therefore, the second person should roll any of the remaining 5 numbers, the third person should roll any of the 4 remaining numbers, and the fourth person should roll any of the 3 remaining numbers.
Thus, we can find the total number of ways such that no two people roll the same number as
\[6 \times 5 \times 4 \times 3 = 360\]
Next, we find the probability that no two people roll the same number.
Let \[E\] be the event that no two people roll the same number.
The total number of possible outcomes is 1296.
The number of favourable outcomes is 360.
Therefore, we get
\[P\left( E \right) = \dfrac{{360}}{{1296}} = \dfrac{5}{{18}}\]
Finally, we will calculate the probability that at least two people roll the same number.
Let \[F\] be the event that at least two people roll the same number.
The events \[E\] and \[F\] are complementary events.
Therefore, we get
\[P\left( F \right) = 1 - P\left( E \right)\]
Substituting \[P\left( E \right) = \dfrac{5}{{18}}\] in the expression, we get
\[\begin{array}{l} \Rightarrow P\left( F \right) = 1 - \dfrac{5}{{18}}\\ \Rightarrow P\left( F \right) = \dfrac{{13}}{{18}}\end{array}\]
\[\therefore\] The probability that at least two people roll the same number is \[\dfrac{{13}}{{18}}\].

Note:
Here we have used the formula of the probability of complementary events. Complementary events means that there are only two events and both are opposite to each other. The sum of the probability of complementary events is always equal to 1. We can also solve this question by finding the ways where two people get the same number, three people get the same number, and all four people get the same number. Then we will add those numbers of ways to get the answer. However, that would be harder to solve.