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Bacteria Puzzle

Q. A jar of water has a single cell of bacteria. With every passing minute, the bacteria will either die, stay the same or divide into two with probability 1/5, 2/5,2/5 respectively. What is the probability that the family of bacteria will survive forever.


A. This is a good example of a puzzle that is most easily solved by assuming a recursive structure. Assume that the probability to extinction is 'x'. Thus the probability that any one bacteria and its descendants will perish is x. The fact that the probability of descendants surviving forever is the same for all generations is key to understanding this problem. This is shown in the visual below.

Let us take the three possible scenarios one at a time.
  1. Bacteria dies (probability = 1/5)
  2. Bacteria stays the same (probability = 2/5)
  3. Bacteria divides into two (probability = 2/5)
We needn't dig further into case 1). The probability is 1/5. For case 2), the  probability that the bacteria would perish is 2x/5. For case 3), that probability works out to be


The sum total of the two should be the same in the first generation as it would be in the second generation. This yields the equation


Solving for x, yields x =19% & x = 100%. We discard the x = 100% solution as we already know that there is 1/5 chance that the bacteria wouldn't survive from the first generation.


If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Here are a few

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

Introduction to Probability Theory

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

Let There Be Range!: Crushing SSNL/MSNL No-Limit Hold'em Games
Easily the most expensive book out there. So if the item above piques your interest and you want to go pro, go for it.

Quantum Poker
Well written and easy to read mathematics. For the Poker beginner.


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.


Comments

  1. Solving for x also yields x = 100%. Thus you should also explain why the other root is the good one.

    ReplyDelete
  2. Thanks 'did' for pointing out that case. I'll update accordingly.

    ReplyDelete
  3. 19% is not a solution for the resulting equation :( it should be 50%

    ReplyDelete

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