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Drawing Aces from a Deck

Q: What is the average number of cards you need to draw from a well shuffled deck of cards before you get an Ace?

The Moscow Puzzles: 359 Mathematical Recreations (Dover Recreational Math)

A:There is a lot of discussion on the web on using hypergeometric distributions for solving these kind of problems. The hypergeometric distribution is just a big word for something fairly simple. Here is the wikipedia explanation for it (link).
But an easier "smarter" way to solve this puzzle (along with ones that fit this framework) is to work with expectations and indicator random variables. Indicator random variables are like on/off switches. They are 1 under certain conditions, 0 otherwise.

Let us assume that the deck of cards (52 total) is as follows, it has 4 aces and all others are labelled X. Let \(Z_i\) represent the indicator variable for a card in position \(i\) with value set as 1 when all aces are in behind it, 0 otherwise.

The total number of draws will then be

$$ N = \sum_{i=1}^{48} Z_{i}$$

Now consider the first draw. The probability that all the aces in the deck is behind it is \(\frac{1}{5}\). This step is crucial to understand. Do you see why it is \(\frac{1}{5}\)? See the figure below. There are 5 scenarios that can play out in terms after the first draw of how the 4 aces position themselves with respect to the current card being pulled AND they are all equally likely.




Only the top row is the one that causes us to draw more cards. This implies that the expectation of \(Z_i\) is always \(\frac{1}{5}\). We run the sum to 48, because after we draw 48 cards there will only be aces left. Thus the sum above yields \(\frac{48}{5} = 9.6\).
Notice when we said that the first row was the only favourable one, it implicitly implied that we are NOT drawing an Ace. Thus the actual count of cards drawn would be 1 + 9.6 = 10.6.

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

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.


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The Three Magical Boxes



Q: You are playing a game wherein you are presented 3 magical boxes. Each box has a set probability of delivering a gold coin when you open it. On a single attempt, you can take the gold coin and close the box. In the next attempt you are free to either open the same box again or pick another box. You have a 100 attempts to open the boxes. You do not know what the win probability is for each of the boxes. What would be a strategy to maximize your returns?

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