STAT 100A HWIII Solution
Problem 1:
Suppose an urn has
r
red balls and
b
blue balls. We random pick a ball, and then we
put three balls of the same color back to the urn. After that we randomly pick a ball again.
(1) What is the probability that the second pick is red?
A:
P
(2nd = red)
=
P
(1st = red)
P
(2nd = red

1st = red) +
P
(1st = blue)
P
(2nd = red

1st = blue)
=
r
r
+
b
r
+ 2
r
+
b
+ 2
+
b
r
+
b
r
r
+
b
+ 2
=
r
r
+
b
.
(2) Optional: Suppose after the second pick, we again put three balls of the same color back
to the urn, and then we randomly pick a ball again. What is the probability that the third pick
is red? (The optional questions will be graded but will not be counted to the total score of your
homework. They are just for fun.)
A:
P
(3nd = red)
=
P
(1st = red)
P
(2nd = red

1st = red)
P
(3rd = red

2nd = red
∩
1st = red)
+
P
(1st = red)
P
(2nd = blue

1st = red)
P
(3rd = red

2nd = blue
∩
1st = red)
+
P
(1st = blue)
P
(2nd = red

1st = blue)
P
(3rd = red

2nd = red
∩
1st = blue)
+
P
(1st = blue)
P
(2nd = blue

1st = blue)
P
(3rd = red

2nd = blue
∩
1st = blue)
=
r
r
+
b
r
+ 2
r
+
b
+ 2
r
+ 4
r
+
b
+ 4
+
r
r
+
b
b
r
+
b
+ 2
r
+ 2
r
+
b
+ 4
+
b
r
+
b
r
r
+
b
+ 2
r
+ 2
r
+
b
+ 4
+
b
r
+
b
b
+ 2
r
+
b
+ 2
r
r
+
b
+ 4
=
r
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 Fall '07
 Wu
 Conditional Probability, Probability, Million, kij

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