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July 16, 2020

How to Pursue Mathematics after High School?

For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.

This is a very beautiful sample problem from ISI MStat PSB 2005 Problem 2 based on finding probability using the binomial distribution. Let's give it a try !!

Problem- ISI MStat PSB 2005 Problem 2


Let \(X\) and \(Y\) be independent random variables with X having a binomial distribution with parameters 5 and \(1 / 2\) and \(Y\) having a binomial distribution with parameters 7 and \(1 / 2 .\) Find the probability that \(|X-Y|\) is even.

Prerequisites


Binomial Distribution

Binomial Expansion

Parity Check

Solution :

Given \( X \sim \) Bin(5,1/2) and \( Y \sim \) Bin(7,1/2) , and they are independent .

Now , we have to find , \( P(|X-Y|=even ) \)

\( |X-Y| \)= even if both X and Y are even or both X and Y are odd .

Therefore \( P(|X-Y|=even )=P(X=even,Y=even) + P(X=odd , Y=odd) \)

P(X=even , Y= even ) =\( ( {5 \choose 0} {(\frac{1}{2})}^5 + {5 \choose 2} {(\frac{1}{2})}^5 + \cdots + {5 \choose 4} {(\frac{1}{2})}^5 )( {7 \choose 0} {(\frac{1}{2})}^7 + {7 \choose 2} {(\frac{1}{2})}^7 + \cdots + {7 \choose 6} {(\frac{1}{2})}^7)\)

=\( ({(\frac{1}{2})}^5 \times \frac{2^5}{2})({(\frac{1}{2})}^7 \times \frac{2^7}{2}) \)

= \(\frac{1}{4} \)

Similarly , one can find P(X=odd , Y=odd ) which is coming out to be \( \frac{1}{4} \) .

Hence , P(|X-Y|) = 14+1/4 = 1/2 .


Food For Thought

Try to find P(X-Y=odd) under the same condition as given in the above problem .


ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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  • What are some of the best colleges for Mathematics that you can aim to apply for after high school?
  • How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
  • What are the best universities for MS, MMath, and Ph.D. Programs in India?
  • What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
  • How can you pursue a Ph.D. in Mathematics outside India?
  • What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

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