Randomly distribute

Probability Level 4

You are a teacher and have $n$ students, they have had an exam recently, and by some miracle all the names on the papers have vanished, the students recognise their own handwriting but as a teacher you don’t, so after correcting all their papers, randomly return it back to them and tell them that if a student has not got his/her paper, he/she should return it back to you. What is the probability all the papers return to you as $n$ tends to $∞ ?$

Give upto five decimal places of the answer.

Note

The answer lies between 0 and 1
You as a teacher have infinite correcting power and can correct papers at light speed.
You can handle an infinite number of students

The answer is 0.36788.

1 solution

Jason Gomez
Mar 2, 2021

If all the students return the paper, it means none of them had got their own paper, so we can use derangements to solve this problem

The number of ways of de-arranging $n$ objects is $D_n$

Where $D_n=n!(\frac 1{0!}-\frac 1{1!} + \frac 1{2!}-\frac 1{3!} … \frac 1{n!})$

As $n$ tends to $∞$ ,using the expansion of $e^x$ , $D_n$ becomes

$D_n=n!(e^{-1})$

Since there are $n!$ ways of giving the children their papers, and all of them are equally probable

The probability that none of them get their own paper is equal to $\frac{D_n}{n!}$

Which equals $\frac 1e ≈ \boxed{0.36788}$ , when $n$ tends to $∞$

If you have used any other methods please put it down here

Jason Gomez - 3 months, 1 week ago

Randomly distribute

The answer is 0.36788.

1 solution

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