Question

Roll three (6-sided) dice. Let X denote the maximum of the values that appear.

a. Find P(X=1).??

b. Find P(X=2).??

c. Find P(X=3).

d. Find P(X=4).??

e. Find P(X=5).?

f. Find P(X=6).

[Hint: It might be helpful to first find the values of P(X?x).]

Answer 1

When three dice are rolled possible number of outcomes are 6*6*6= 216.

(a)

Out of 216 outcomes, one outcome has all 1 so

P(X=1) = 1/216

(b)

The maximum 2 can be possible if at least one die shows 2 and rest dice shows 1.

That is outcome can have all 2's (in one way) or 2 2s and 1 1s (in three ways) or 1 2s and 2 1s (in three ways).

The possible number of outcomes with maximum 2: 1+3+3 = 7

Therefore

P(X=2) = 7/216

(c)

The maximum 3 can be possible if at least one die shows 3 and rest dice shows 1 or 2.

The possible number of outcomes are: 19

P(X=3) = 19/216

(d)

The maximum 4 can be possible if at least one die shows 4 and rest dice shows 1 or 2 or 3.

The possible number of outcomes are: 37

P(X=4) = 37/216

(e)

The maximum 5 can be possible if at least one die shows 5 and rest dice shows 1 or 2 or 3 or 4.

The possible number of outcomes are: 61

P(X=5) = 61/216

(f)

The maximum 6 can be possible if at least one die shows 6 and rest dice shows 1 or 2 or 3 or 4 or 5.

The possible number of outcomes are: 91

P(X=6) = 91/216