Unformatted text preview: Example 2.10.2 Consider a two stage experiment. First roll a die, and let X denote the number
showing. Then ﬂip a fair coin X times, and let Y denote the total number of times heads shows.
Find P {Y = 3} and P (X = 3Y = 3).
Solution: By the law of total probability,
6 P {Y = 3} = P (Y = 3X = j )pX (j )
j =1 =
=
= 1
0+0+
6
1
0+0+
6
1
.
6 3 −3
4 −4
5 −5
6 −6
2+
2+
2+
2
3
3
3
3
1 1 10 20
++
+
8 4 32 64 50 CHAPTER 2. DISCRETETYPE RANDOM VARIABLES Now P {X = 3, Y = 3} is the term in the above sum for j = 3: P {X = 3, Y = 3} =
Therefore,
P {X = 3, Y = 3}
1
P (X = 3Y = 3) =
=.
P {Y = 3}
8 2−3
6 = 1
48 . Example 2.10.3 Consider two boxes as shown in Figure 2.9. Box 1 has three black and two white Box 1 Box 2 Figure 2.9: Initial state of boxes.
balls, while Box 2 has two black and two white balls. Consider the following two step experiment.
Step 1: Select a ball from Box 1, all ﬁve having equal probability, and transfer it to Box 2.
Step 2: Remove a ball from Box 2, with each of the ﬁve balls having the same chance of being
removed.
Deﬁne the following two events:
W=“the ball that is transferred is white...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.
 Spring '08
 Zahrn
 The Land

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