Get inspired by the success stories of our students in IIT JAM 2021. Learn More 

June 2, 2020

ISI MStat 2018 PSA Problem 13 | Probability of functions

This is a beautiful problem from ISI MStat 2018 PSA problem 13 based on probability of functions. We provide sequential hints so that you can try .

Probability - ISI MStat Year 2018 PSA Problem 13


Consider the set of all functions from \( {1,2, \ldots, m} \) to \( {1,2, \ldots, n} \) where \( n>m .\) If a function is chosen from this set at random, what is the probability that it will be strictly increasing?

  • \( {n \choose m} / m^{n} \)
  • \( {n \choose m} / n^{m} \)
  • \( {{m+n-1} \choose m} / m^{n} \)
  • \( {{m+n-1} \choose m-1} / n^{m} \)

Key Concepts


combination

Check the Answer


Answer: is \( {n \choose m} / n^{m} \)

ISI MStat 2018 PSA Problem 13

A First Course in Probability by Sheldon Ross

Try with Hints


What is the total number of functions from \({1,2, \ldots, m}\) to \({1,2, \ldots, n}\) where (n>m)

You have to choose \(m\) numbers among \({1,2, \ldots, n}\) and assign it to the \({1,2, \ldots, m}\)
For each element of \({1,2, \ldots, m}\), there are \(n\) options from \({1,2, \ldots, n}\).
Hence \(n^m \) number of functions .

\(f(i) = a_i \)

The number of ways to select Select \(m\) elements among \({1,2, \ldots, n}\) is the same as the number of strictly ascending subsequences of length m taken from 1, 2, 3, ..., n, which is the same as the number of subsets of size m taken from {1,2,3,…,n}, which is \( {n \choose m} \) .

Hence the probability that it will be strictly increasing \( \frac{ {n \choose m} }{ n^{m} } \)

Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com