Question

In: Statistics and Probability

3. Let the experiment be the toss of three dice in a row. Let X be...

3. Let the experiment be the toss of three dice in a row. Let X be the outcome of the first die. Let Y be the outcome of the 2nd die. Let Z be the outcome of the 3rd die. Let A be the event that X > Y , let B be the event that Y > Z, let C be the event that Z > X.

(a) Find P(A).

(b) Find P(B).

(d) Are A and B independent?

(e) Are A, B, C pairwise independent?

(f) Find P(A ∩ B ∩ C).

(g) Are A, B, C mutually independent?

Solutions

Expert Solution

In this experiment the total number of outcomes are .

(a) The event A says X>Y but it doesn't put any conditions on Z. So all values of Z are considered.

So for X=x there will be (x-1) possible values of Y in the set {6,5,4,3,2,1}, where x=2,3,4,5,6.

And for every chosen pair (x,y) there are 6 possible values of Z.

So the number of favourable outcomes is

(b) The problem is similar as (a) just instead of X>Y, we have Y>Z and here X can take any of 6 possible values.

So the required probability is same as above, i.e. .

(c) Now

So for X=x, Y can be chosen from the set {2,3,...,(x-1)} and for Y=y there will be (y-1) possible values of Y in the set {6,5,4,3,2,1}, where x=3,4,5,6 and y=2,3,...,(x-1).

So the number of favourable outcomes is

(d) Since , A and B are not independent.

(e) Since the cases for and are similar to the case of , the results will be same.

So B and C are not independent and A and C are not independent.

So A,B,C are not pairwise independent.

(f) Now,

So this is an impossible event Since X>Z and Z>X can't happen at same time.

(g)

Since A,B,C are not mutually independent.

orchestra answered 1 year ago

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