Q: A & B agree to stake $100 each for a coin tossing game involving 10 tosses of a coin. Whoever gets more heads wins. However, the coin is made of glass and at the 7th toss, when A has won 4 heads and B has won 3 heads, it breaks. How should the total of $200 be divided in a fair manner between the two?

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A: A fair way to proceed is to give A all the credit aggregated so far and do the same for B with the obvious intuition being that A deserves more as he was closer to win than B. But how does one compute that? On any given toss, the probability that either would win is \(\frac{1}{2}\). Without loss of generality, it would be fair to say that had the tossing not been stopped, A would have won if he won two or more times. Being disjoint events, this probability works out to

$$

\big(\frac{1}{2})^{3} + \big(\frac{1}{2})^{3} = \frac{1}{4}

$$

Likewise, B would have won if he won all three tosses in a row. This happens with a probability \(\big(\frac{1}{2})^{3} = \frac{1}{8}\). Thus, the probability of A winning can be normalized as

$$

P(\text{A wins}) = \frac{\big(\frac{1}{4})}{\big(\frac{1}{4}) + \big(\frac{1}{8})} = \frac{2}{3}

$$

This fraction is also a fair way to split the pot of money.

Some good books to learn the art of probability

Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)

This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.

Introduction to Algorithms

This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists

An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)

Introduction to Probability, 2nd Edition

The Mathematics of Poker

Good read. Overall Poker/Blackjack type card games are a good way to get introduced to probability theory

Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)

An excellent resource (students/engineers/entrepreneurs) if you are looking for some code that you can take and implement directly on the job.

Understanding Probability: Chance Rules in Everyday Life A bit pricy when compared to the first one, but I like the look and feel of the text used. It is simple to read and understand which is vital especially if you are trying to get into the subject

Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems) This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.

Discovering Statistics Using R

This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

can you please explain why prob(A would have won)=1/4

ReplyDeletei calculated it as 4/8

for the cases AAA AAB ABA BAA