Probability Shortcut: 3 Dices Rolled Together

Published on Friday, July 24, 2015
Today I am going to share Probability shortcuts for problems based on Dice. Must Probability trick: When 2 Dices rolled together

Sum of dices when three dices are rolled together
If 1 appears on the first dice, 1 on the second dice and 1 on the third dice.
         
(1, 1, 1) = 1+1+1=3
The minimum sum with three dices rolled together = 3
If 6 appears on the first dice, 6 on the second dice and 6 on the third dice.
        
(6, 6, 6) = 6+6+6 =18
   
The maximum sum with three dices rolled together = 18
The sum could be 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18.

  • What is the probability of getting a sum of 4 if three dices are rolled together.
        Solution:   
dice problems
            Total outcomes = 6x6x6 =216


Required probability = favorable case / total outcomes
                                                    = 3/216
                                                    = 1/72

  • What is the probability of getting a sum of 5 if you throw three dices together?
         Solution:
             
                                   
dice problems
                             Required Probability =  6/216 = 1/36





Now as you can see that it is a time consuming method, let us find a quick method to find the probability.


There are total 16 numbers (possible number of sums) and half of 16 is 8.

Counting 8 from both the sides we get 10 and 11 in the middle.

probability shortcut

Now we will construct the triangle which would help us to find the probability if three dices are rolled together.
probability trick
Memorizing the making of the above picture makes the 

calculation easier and faster.

  • What is the probability of getting a sum of 11 if three dices are rolled together?
         Solution:
                               From the table

                               Sum 11 = probability 27

                               Required probability = 27/216 =1/8 

  • What is the probability of getting a sum of 14 if three dices are rolled together?
         Solution:
                                     From the table

                                    Sum 14 = probability 15

                                    Required probability = 15/216

  • What is the probability of getting a sum of 11 or 9 if three dices are rolled together?
Solution:

From the table

Sum 11 = probability 27   and   sum 9 = probability 25

Since there is an ‘and’ so we will add

Required Probability = 27/216 +25/216 = 52/216

  • If 3 dices are rolled at the same time. What is the probability of all 3 dice showing even face values?
Solution:  Even faces means 2, 4, 6

There are 3 ways 2 can appear on in first go.

Similarly, there are 3 ways 4 can appear on in first go.
.
There are 3 ways 6 can appear on in first go.

So total number of ways: 3+3+3

Required probability = 9/216 =1/24

  • What is the probability of not getting 6 when 3 dice are rolled together?
Solution:

Probability of not getting 6 on the first die = 5/6

(As probability of getting 6 on first die is 1 so the probability of not getting 6 = 6-1=5)

And

Similarly, Probability of not getting 6 on the second die = 5/6

And

Probability of not getting 6 on the third die = 5/6

So the required probability = 5x5x5/216=125/216

Remember AND means multiply and OR means ADD.



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