Posts | Rishi

The Asymmetric Coin Toss Problem

Sat, 22 Jul 2023 00:00:00 +0000

As promised I am back with a new problem from the collection. For new readers, I have recently started a new series where I bring interesting probability puzzles and their solutions to present some revealing aspect of our thought process — something which I really wanted when I learnt about Probability (someone who can walk me through each step and guide why some way of thinking was wrong). You can also contribute your problems or solutions to this series, by reaching me out on LinkedIn, Twitter, Instagram or Email.

The problem goes like this…
Two gamblers are playing a coin toss game. Gambler A has (n + 1) fair coins; B has n fair coins. What is the probability that A will have more heads than B if both flip all their coins?

Read my detailed solution at: The Asymmetric Coin Toss Problem

The Successive Wins Problem

Thu, 20 Jul 2023 00:00:00 +0000

I fortuitously ran into one of my old friend, who instigated my interest in probability puzzles once again. This time not just with a problem but an entire collection of it. So, I decided to devote some time and effort to create a series of blog posts to bring them to you accompanied by their solution. If you are interested about this series follow me to stay tuned. You can also contribute your problems or solutions to this series, by reaching me out on LinkedIn, Twitter, Instagram or Email.

The problem goes like this…
To encourage Elmer’s promising tennis career, his father offers him a prize if he wins (at least) two tennis sets in a row in a three-set series to be played with his father and the club champion alternately: father-champion-father (FCF) or champion-father-champion (CFC), according to Elmer’s choice. The champion is a better player than Elmer’s father. Which series should Elmer choose? Assume that Elmer stops playing when he has satisfied the winning rule or cannot satisfy the winning rule anymore.

Read my detailed solution at: The Successive Wins Problem

Kernel Density Estimation — A Gentle Introduction to Non-Parametric Statistics

Fri, 03 Feb 2023 00:00:00 +0000

Normality is a Myth!

Back in the 20th century, when Statistics was still in its infancy and computers weren’t that popular, it was norm to assume normality as the distribution from which data was generated. It was mostly because it made the calculations less tedious in the age when all results were hand calculated.

But with the rise of computational power these assumptions can safely be put aside and more insights can be drawn directly for the data. Even the availability of data in this Big Data era made Statisticians to adopt more modern techniques — Non-Parametric Statistics. Here we will discuss one such method to estimate the probability distribution, Kernel Density Estimation.

Revisiting Logistic Regression — A Gentle Introduction to Generalized Linear Models

Fri, 03 Feb 2023 00:00:00 +0000

All models are wrong, but some are useful

The two fundamental pillars of supervised statistical learning — Regression and Classification. Simple Linear Regression and Logistic Regression is how many of us have started our journey in Statistics and Data Science. A long-standing debate still prevails on why is Logistic Regression a Classification Model instead of a Regression Model?

Here we revisit Logistic Regression from an intuitive perspective along with statistical rigor. We will briefly touch up on the concepts behind Generalized Linear Model along with an optional section on Iterative Re-weighted Least Squares (IRLS) to fit these models.

The Three Musketeers in a Semicircle Problem

Sat, 29 Oct 2022 00:00:00 +0000

I came across this interesting problem on which my seniors were brainstorming late at night. It requires a bit knowledge of continuous random variables in probability and little bit visualization capability.

The problem goes like this…
There are three musketeers who stand on a circle randomly, which follows Uniform(0, 2πr) independently where r is the radius of the circle. They can help each other if they are close enough in case of a sudden attack by the enemies. Find the probability of them lying on the same semicircle.

Read my detailed solution at: The Three Musketeers in a Semicircle Problem

The Rabbit and the Stone Problem

Thu, 27 Oct 2022 00:00:00 +0000

I have this friend of mine, who constantly pokes me to solve interesting puzzles. He sometimes even rings me up, just to look at the problem he sent :P This is one of those problems, that I enjoyed working on. It requires understanding of basic probability theory, counting and some time to stick to a problem that interests you.

The problem goes like this…
There are 7 stones in a row labelled 1 to 7 from left to right. A rabbit starts at stone 1 and on each turn hops to a new stone that they haven’t visited yet. After visiting all the stones, what is the probability the rabbit made exactly one leftwards jump?

Read my detailed solution at: The Rabbit And The Stone Problem

The Classic Urn Color Change Problem

Sat, 22 Oct 2022 00:00:00 +0000

Anyone who has taken probability class in their college or school is familiar with probability problems involving the classic urn (basically a bag) setup which contains balls of different colors and the professor asks you to find the probability of occurrence of some stupid pattern. This is one of those problems.

The problem goes like this…
You are given an urn with 100 balls (50 black and 50 white). You pick balls from urn one by one without replacements until all the balls are out. A black followed by a white or a white followed by a black is “a color change”. Calculate the expected number of color changes if the balls are being picked randomly from the urn.

Read my detailed solution at: The Classic Urn Color Change Problem

The Chessboard High Five Problem

Fri, 21 Oct 2022 00:00:00 +0000

Just another day, when I was browsing through my Instagram feed, my friend sent me this interesting yet not so difficult problem involving understanding of basic probability, counting and bit of time to pause the temptation to scroll the rest of your Instagram feed.

So, the problem went like this..
Imagine you have a regular 8 x 8 chessboard, with 64 people standing on each of the 64 squares. At the same time every person turns to face one of their neighbors with equal chances i.e. say the person near the center of the chessboard can face any direction with 1/4 probability, a person at the top of the chessboard (but not the corners) has probability 1/3 to face any of its neighbors, similarly all others. Now, if two people are facing each other they give each other a high five. Find the expected number of high fives.

Read my detailed solution at: The Chessboard High Five Problem