In: Statistics and Probability
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).]
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