Randomly Uniformly!

Probability Level 5

You are given $n$ urns (numbered 1 to n) with each containing $n-1$ balls such that ${i}^{th}$ urn contains $i-1$ blue balls and $n-i$ red balls.

You perform the following set of events in sequence.

Choose one urn randomly uniformly and then pick up one ball randomly uniformly from it and throw it away.
Choose one urn randomly uniformly and then pick up 2 balls randomly uniformly from it and throw them away.
Choose one urn randomly uniformly and then pick up 3 balls randomly uniformly from it and throw them away.
Choose one urn randomly uniformly and then pick up 4 balls randomly uniformly from it and throw them away.
Choose one urn randomly uniformly and then pick up 5 balls randomly uniformly from it and number them from 1 to 5 randomly uniformly.

What is the probability that the ball you numbered 5 is blue ? ( Take n = 2018 if needed ).

Enter answer upto 5 decimal places.

Original :)

The answer is 0.50000.

2 solutions

Pranav Rao
Sep 25, 2018

The probability that the 5th ball in the last round is blue is just the number of required events, divided by the total number of events. We can show a bijection between the number of events where 5th ball in the last round is blue and the events where 5th ball in the last round is red: If the urns chosen in the event are $i_{1}$ , $i_{2}$ , $i_{3}$ , $i_{4}$ , $i_{5}$ , then choose urns $n - i_{1}$ , $n - i_{2}$ , $n - i_{3}$ , $n - i_{4}$ , $n - i_{5}$ and then just change the color of the balls picked each time.

Aniket Sanghi
Sep 20, 2018

Its pure Randomness.

The probability is simply the probability of selecting a blue ball from the given urns. And since the no. of blue balls and red balls are equal and you are choosing balls as well as the urns randomly uniformly the probability of getting a blue ball is just half!

Think over it!

Could you lay out how many balls (blue and red) are in the first few urns, as I'm not following your question nor solution. My understanding would be there is a conflict in the initial statement you've made.

David Entwistle - 2 years, 8 months ago

n is some fixed value. For n=2018, there are 2018 urns each having 2017 balls. such that 1st has all red balls, second has one blue and 2016 red.... similarly 2018th urn contains all blue balls

Aniket Sanghi - 2 years, 8 months ago

The reasoning is incorrect. You're claiming that the last ball is being picked uniformly from all possible balls, but this doesn't follow: an urn with less balls have its balls more likely to be picked. (Of course, it also means there was a possibility that those balls got removed too, and if I worked it out correctly the probabilities do cancel exactly, but you have to take that into account in your solution.)

Ivan Koswara - 2 years, 7 months ago

Think in the other way.... If you do 2 different process one after the other randomly uniformly without knowing the outcome of any. The final result is independent of which process you do first.

Aniket Sanghi - 2 years, 7 months ago

Randomly Uniformly!

The answer is 0.50000.

2 solutions

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