1 . Consider a sphere of radiusrcentered at (0,0,0). Express the following sets in set notation.

(a) The points on the surface of the sphere.

(b) The points inside the sphere.

(c) The points outside the sphere.

(d) The points inside the sphere but outside the sphere of radiusr/2centered at (0,0,0).

(e) The points inside the sphere but outside the cube of length3r/2centered at (0,0,0).

2 . Suppose we have these points:

A= (a1,a2),B= (b1,b2)

C= (c1,c2), D= (d1,d2)

where a1< b1< c1< d1and a2< b2< c2< d2.

NOTE: The path only move right or up on the grid.

(a)give an expression for the probability that a random path fromAtoDwill pass through bothBandC.

(b)give an expression for the probability that a random path fromAtoDwill pass through B, but notC.

(c)give an expression for the probability that a random path fromAtoDwill pass through neitherBnorC.

3 . Find the probability mass function of a random variableXcorresponding to the absolute value of the difference between the results of two dice.

4 . Suppose 10 fair coins are tossed. Find the probability there will be more heads than tails.

5 . Suppose that a woman is equally likely to have a male or female child. A couple wishes to have exactly three female children in their family. They will have children until this condition is fulfilled.

(a) What is the probability that the family haskmale children?

(b) What is the probability that the family has four children?

(c) What is the probability that the family has at most four children?

6 . Suppose Archana and Katie play a game where they alternate (in this order) in rolling two fair dice until the first person rolls a sum of 8 on a single turn, who wins the game.

(a)Find the probability that each person wins the game.

(b)How much of an advantage does Archana have?

(c)Suppose Archana goes first, then Katie getskturns, and then they alternate. How large mustkbe so that Katie is more likely to win?