Question

Four fair dices were rolled.

Part(a) How many possible outcomes there will be, if the order of dices are considered and their faces (number of points) are recorded?

Part(b) How many possible outcomes there will be, if the sum of the points of the four dices are recorded?

Part (c) What is the probability of getting a result with the sum exactly equals to 6?

Part(d) What is the probability of getting a result with the sum no less than 6?

Answer 1

Part(a)

Since, we have 4 dices and each dice can have 6 outcomes.

So, Number of Possible outcomes if the order of dices are considered and their faces (number of points) are recorded

= 6⁴ = 1296

Part(b)

If sum of four dices are recorded, then the least value we can get is 4 ( which means 1 on each dice) and maximum value we can get is 24 ( which means 6 on each dice).

So, we can get each and every value starting from 4 till 24 as a sum of the points of the four dices.

So, in total , possible outcomes for the sum of the points of the four dices = 21

Part (c)

So, let us consider the cases when a result with the sum exactly equals to 6 is observed.

Favorable cases and there counts are :

Combo	Count
1-1-1-3	4!/3! = 4
1-1-2-2	4!/(2!*2!) = 6
Total	10

So, in total there are 10 cases when a result with the sum exactly equals to 6 is observed.

Probability of getting a result with the sum exactly equals to 6 =

= Number of cases whensum exactly equals to 6 is observed / Total cases

= 10 / 1296

= 0.0077

Part(d)

Probability of getting a result with the sum no less than 6 = 1 - Probability of getting a result with the sum less than 6

= 1 - [ Probability of getting a result with the sum equal to 4 +  Probability of getting a result with the sum equal to 5 ]

Consider the cases when a result with the sum exactly equals to 4 is observed.

Favorable cases and there counts are :

Combo	Count
1-1-1-1	4!/4! = 1
Total	1

Consider the cases when a result with the sum exactly equals to 5 is observed.

Favorable cases and there counts are :

Combo	Count
1-1-1-2	4!/3! = 4
Total	4

Probability of getting a result with the sum equal to 4 = 1 / 1296 = 0.0008

Probability of getting a result with the sum equal to 5 = 4 / 1296 = 0.0031

Probability of getting a result with the sum no less than 6 = 1 - ( 0.0008 + 0.0031) = 0.9961