a pluduet pPomotion (with a patriotic theme) for the month of July, a soft drink produces glass bottles of soda with the old-fashioned metal cap. Printed on the insidec cap is the name of a US state or territory, and a scene from that state or territory. If or accumulate a complete set of all N different bottle caps, then one can turn the set in ar prize. Assume each state or territory appears at random with an equal probability; also a no trading with friends. (There was actually was a contest like this when I was a small boy. For that contest, N w states, plus Puerto Rico, the District of Columbia, the US Virgin Islands, Guam, and Am- but we consider the general case.) (a) ind
a pluduet pPomotion (with a patriotic theme) for the month of July, a soft drink produces glass bottles of soda with the old-fashioned metal cap. Printed on the insidec cap is the name of a US state or territory, and a scene from that state or territory. If or accumulate a complete set of all N different bottle caps, then one can turn the set in ar prize. Assume each state or territory appears at random with an equal probability; also a no trading with friends. (There was actually was a contest like this when I was a small boy. For that contest, N w states, plus Puerto Rico, the District of Columbia, the US Virgin Islands, Guam, and Am- but we consider the general case.) (a) ind
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Question
Transcribed Image Text:4. As a product promotion (with a patriotic theme) for the month of July, a soft drink manufacturer
produces glass bottles of soda with the old-fashioned metal cap. Printed on the inside of each bottle
cap is the name of a US state or territory, and a scene from that state or territory. If one manages to
accumulate a complete set of all N different bottle caps, then one can turn the set in and win a nifty
prize. Assume each state or territory appears at random with an equal probability; also assume there's
no trading with friends.
(There was actually was a contest like this when I was a small boy. For that contest, N was 55: the 50
states, plus Puerto Rico, the District of Columbia, the US Virgin Islands, Guam, and American Samoa;
but we consider the general case.)
(a) Find the probability that one has a complete set of bottle caps after purchasing N sodas.
(b) What is the expected value of the number of sodas one must purchase to obtain a complete set.
(c) What is variance of the number of sodas one must purchase to obtain a complete set.
Solution Remarks
Let the random variable X, be the number of sodas it takes to get from having i –1 different bottle caps
to i different bottle caps. The number of sodas, W, one needs to get a complete set of bottle caps is:
W = X1 + X2 + +XN
Note each of the X is a geometric random variable. ( Different parameter p for each i.) Also, one can
argue that the X;'s are independent when obtaining the variance of W,
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON