1Assume that we have a tetrahedral die with the four values

1)Assume that we have a tetrahedral die with the four values 1, 2, 3 and 4 on the die that is rolled 3 times.

a)What is the probability that all 4 values will be the same?

b)What is the probability that all 4 values will be the different?

c)What is the probability that all exactly 2 of the values will be the same?

Solution

a)

Only 4 cases are possible

All 4 values are 1 or all 4 are 2 or all 4 are 3 or all four are 4

But total possible cases are:4*4*4*4

So, 4/4^4=1/4^3=1/64

b)

All 4 values are possible on first dice and then 3 values are possible on second die since all four dice must have different values., and then 2 on the third dice and 1 on the last dice

So, 4*3*2 cases

So probability,p=(4*3*2)/(4^4)=6/64=3/32

c)

First two dice can have 4 possible cases:

Both 1s ,both 2s ,both 3s or both 4s

Other two dies must both not have the values on first two dies

So 3 values possible on each.

So,4*3*3 cases

So probability,p=(4*3*3)/4^4=9/64