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An urn initially contains one red and one blue ball. At
Chapter 4, Problem 35TE(choose chapter or problem)
Problem 35TE
An urn initially contains one red and one blue ball. At each stage, a ball is randomly chosen and then replaced along with another of the same color. Let X denote the selection number of the first chosen ball that is blue. For instance, if the first selection is red and the second blue, then X is equal to 2.
(a) Find P{X > i},i ≥ 1.
(b) Show that with probability 1, a blue ball is eventually chosen. (That is, show that P{X<∞} = 1.)
(c) Find E[X].
Questions & Answers
QUESTION:
Problem 35TE
An urn initially contains one red and one blue ball. At each stage, a ball is randomly chosen and then replaced along with another of the same color. Let X denote the selection number of the first chosen ball that is blue. For instance, if the first selection is red and the second blue, then X is equal to 2.
(a) Find P{X > i},i ≥ 1.
(b) Show that with probability 1, a blue ball is eventually chosen. (That is, show that P{X<∞} = 1.)
(c) Find E[X].
ANSWER:
Solution:
Step 1 of 3:
Let X = the selection number of the first chosen ball that is blue.
If the first selection is red and the second is blue, then X is equal to 2.
-
The claim is to find P{X > i},
if X = 1, then first draw is blue
P(X > i) = 1 - P(X = 1) - P(X < i)
= 1 - - 0.
= .
if X = 2, then the first draw is red
P(X = 2) = (
)
Then, P(X > 2) = 1 - P(X = 2) - P(X < 2)
= 1 - (
) -
= 1 -
=
Therefore, P(X > i) = (
).....(
)
=
Hence, P{X > i} = .