Probability for Rolling Three Dice

Probability means possibility. It deals with the occurrence of a random event. The example is if you toss a coin, the result will be head or tail. In this case, we use the probability method. Check the following sections to know what is the probability of rolling three dice, solved questions.

Probability for Rolling Three Dice

To find the probability of any event, we have to know the number of favorable outcomes and the total number of outcomes. If you throw a die, the sample space contains numbers from 1 to 6 and the total number of outcomes is 6. In the same way, when three dice are rolled simultaneously the probability becomes difficult. The total number of outcomes in rolling 3 dice is 6³ = 216.

The faces of a die are {1, 2, 3, 4, 5, 6}. You can check the probabilities of rolling three dice example questions with solutions.

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Worked-out problems on 3 Dice Rolling Probability

Problem 1:
Three dice are rolled. What is the probability that the numbers shown are different?

Solution:
The total number of outcomes = 6 x 6 x 6 = 216
Three dice shows different numbers means, if the first dice shows 2, 2nd dice shows 3, 3rd dice shows 4.
The total number of favorable outcomes = 6 x 5 x 4 = 120
P(numbers shown are different) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 120 }{ 216 } \)
= \(\frac { 5 }{ 9 } \)
Therefore, the probability that the numbers shown are different is \(\frac { 5 }{ 9 } \).

Problem 2:
Three dice are thrown together. Find the probability of
(i) getting a sum of 6
(ii) getiing a total of 5
(iii) getting a total of atmost of 6

Solution:
Three different dice are thrown at the same time.
Total no of possible outcomes = 6³ = 216
(i) getting a sum of 6
No of events of getting a total of 6 = 10
Possibilities of getting a sum of 6 = {(1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) and (2, 2, 2)}
P(getting a sum of 6) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 10 }{ 216 } \)
= \(\frac { 5 }{ 108 } \)
(ii) getiing a total of 5
No of events of getting a total of 5 = 6
Possibilities of getting a sum of 5 = {(1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1), (2, 1, 2) and (1, 2, 2)}
P(getting a total of 5) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 6 }{ 216 } \)
= \(\frac { 1 }{ 36 } \)
(iii) getting a total of atmost of 6
No of events of getting a total of atmost 6 = 20
Possibilities of getting a total of atmost 6 = {(1, 1, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 1, 3), (1, 3, 1), (3, 1, 1), (2, 2, 1), (1, 2, 2), (1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) and (2, 2, 2)}
P(getting a total of atmost 6) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 20 }{ 216 } \)
= \(\frac { 5 }{ 54 } \)

FAQ’s on Probability of Rolling Three Dice

1. What is the probability of getting 3 sixes when you roll 3 dice?

The number of possible outcomes is 216 when you roll 3 dice.

The number of possibilities to get 3 sixes is 1

Therefore, P(getting 3 sixes) = \(\frac { 1 }{ 216 } \)

2. How to find the probability of rolling multiple dice?

The simple formula to get the probability of rolling multiple dice are \(\frac { number of desired outcomes }{ number of possible outcomes } \).

3. What is the probability of getting at least one six when you roll 3 dice?

The number of possible outcomes = 216

The number of possibilities of getting at least one six = 5 x 5 x 5 = 125

P(getting at least one six) = \(\frac { 125 }{ 216 } \).

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