## How to roll a Dice by tossing a Coin ? Cheenta Statistics Department

How can you roll a dice by tossing a coin? Can you use your probability knowledge? Use your conditioning skills.

Suppose, you have gone to a picnic with your friends. You have planned to play the physical version of the Snake and Ladder game. You found out that you have lost your dice.

The shit just became real!

Now, you have an unbiased coin in your wallet / purse. You know Probability.

### Aapna Time Aayega

starts playing in the background. :p

## Can you simulate the dice from the coin?

Ofcourse, you know chances better than others. :3

Take a coin.

Toss it 3 times. Record the outcomes.

HHH = Number 1

HHT = Number 2

HTH = Number 3

HTT = Number 4

THH = Number 5

THT = Number 6

TTH = Reject it, don’t ccount the toss and toss again

TTT = Reject it, don’t ccount the toss and toss again

Voila done!

What is the probability of HHH in this experiment?

Let X be the outcome in the restricted experiment as shown.

How is this experiment is different from the actual experiment?

This experiment is conditioning on the event A = {HHH, HHT, HTH, HTT, THH, THT}.

$P( X = HHH) = P (X = HHH | X \in A ) = \frac{P (X = HHH)}{P (X \in A)} = \frac{1}{6}$

Beautiful right?

Can you generalize this idea?

## Food for thought

• Give an algorithm to simulate any conditional probability.
• Give an algorithm to simulate any event with probability $\frac{m}{2^k}$, where $m \leq 2^k$.
• Give an algorithm to simulate any event with probability $\frac{m}{2^k}$, where $n \leq 2^k$.
• Give an algorithm to simulate any event with probability $\frac{m}{n}$, where $m \leq n \leq 2^k$ using conditional probability.

## Watch the Video here:

Books for ISI MStat Entrance Exam

How to Prepare for ISI MStat Entrance Exam

ISI MStat and IIT JAM Stat Problems and Solutions

Cheenta Statistics Program for ISI MStat and IIT JAM Stat

Simple Linear Regression – Playlist on YouTube

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## Circumference of a Semicircle | AMC 8, 2014 | Problem 25

Try this beautiful problem from AMC-8-2014 (Geometry) based on Circumference of a Semicircle

## Circumference of a Semicircle- AMC 8, 2014 – Problem 25

On A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5miles per hour, how many hours will it take to cover the one-mile stretch?

• $\frac{\pi}{11}$
• $\frac{\pi}{10}$
• $\frac{\pi}{5}$

### Key Concepts

Geometry

Semicircle

Distance

Answer:$\frac{\pi}{10}$

AMC-8, 2014 problem 25

Challenges and Thrills of Pre College Mathematics

## Try with Hints

Find the circumference of a semi-circle

Can you now finish the problem ……….

If Robert rides in a straight line, it will take him $\frac {1}{5}$ hours

can you finish the problem……..

If Robert rides in a straight line, it will take him $\frac {1}{5}$ hours. When riding in semicircles, let the radius of the semicircle r, then the circumference of a semicircle is ${\pi r}$. The ratio of the circumference of the semicircle to its diameter is $\frac {\pi}{2}$. so the time Robert takes is  $\frac{1}{5} \times \frac{\pi}{2}$. which is equal to $\frac{\pi}{10}$

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## Spiral Similarity

Geometric Transformation is a powerful tool in Geometry. We look at one such transformation: Spiral Similarity.

## Try these Spiral Similarity problems. Send answers to helpdesk@cheenta.com

Problem 1: Suppose O = (0, 0), A = (2, 0) and B = (0, 2). Let T be the spiral similarity that sends A to B (center of T is O). What is the angle of spiral similarity? What is the dilation coefficient?

Problem 2: Can you rigorously prove the claim made at the end of the video?

Problem 3: Revisit the spiral similarity T described in problem 1. This can be realized by multiplying a complex number to A. What is that complex number?

Also try the problems related to Cyclic Quadrilaterals.

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## Using auxiliary polynomials in ISI Entrance

Auxiliary polynomials are useful for solving complicated polynomial problems. This problem from ISI Entrance (and a Soviet Olympiad) is a good example.

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## Really understanding Direct Limit, Inverse Limit, and Hom

Direct Limit, Inverse Limit, and Hom are three ideas from category theory that are useful in many branches of mathematics. A deep understanding of them can be very helpful in the long run.

In the following video, we draw schematic pictures and gain real intuition behind these abstract ideas. This is clearly one of the most important videos of our production.

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## What is Inversive Geometry?

Inversion is one of several geometric transformations. It has a curious power of converting lines into circles. Inversion can be used to solve many geometry problems. It also has deep relations with complex numbers.