Chapter 15.3, Problem 1P

(a) Set up a sample space for the 5 black and 10 white balls in a box discussed above assuming the first ball is not replaced. Suggestions: Number the balls, say 1 to 5 for black and 6 to 15 for white. Then the sample points form an array something like ( 2.4), but the point 3,3 for example is not allowed. (Why? What other points are not allowed?) You might find it helpful to write the numbers for black balls and the numbers for white balls in different colors.

(b) Let $A$ be the event "first ball is white" and $B$ be the event "second ball is black." Circle the region of your sample space containing points favorable to $A$ and mark this region $A$.

Similarly, circle and mark region $B$.

Count the number of sample points in $A$ and in $B$; these are $N(A)$ and $N(B).$ The region AB is the region inside both $A$ and B; the number of points in this region is $N(AB).$ Use the numbers you have found to verify (3.2) and $(3.1).$ Also find $P(B)$ and $P_B(A)$ and verify (3.3) numerically.

(c) Use Figure 3.1 and the ideas of part (b) to prove (3.3) in general.