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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} }

Similar Problems and Solutions



Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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} }

Similar Problems and Solutions



Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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