Probability of Increasing Sequence

Slips of paper with the numbers from 1 to 99 are placed in a hat. Five numbers are randomly drawn out of the hat one at a time (without replacement). What is the probability that the numbers are chosen in increasing order?

Reference

  1. MIT Opencourseware, PROBLEMS ON "HIDDEN" INDEPENDENCE AND UNIFORMITY, 18.S34 (FALL 2007)

Answer

Solution

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Copyright © 1996-2017 Alexander Bogomolny

1/120. You can check the solution.

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Copyright © 1996-2017 Alexander Bogomolny

It does not at all matter whether there are 99 slips of paper or just 5. The best way to approach the problem is to consider drawing just two numbers. So you've drawn two distinct numbers: what is the probability that the first one will be less than the second? Well, there are just two ways to draw two chosen numbers: the smallest either come first or last, meaning that the probability of drawing two numbers in their natural order is 1/2.

When it comes to five numbers. There are 120 permutations of five objects and thus 120 ways to draw a given set of five numbers. Only in one permutation out of 120 all five numbers follow in their natural order of magnitude.

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Copyright © 1996-2017 Alexander Bogomolny

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