[Guest]

Recently, my boss hired a female secretary, who looks great but matches him when it comes to wits.

Now, yesterday he was dictating a few letters to her. All that the lass had to do was to post them. N letters, N envelopes, no confusion. Or so you will think. 'Coz she randomly puts a letter in each of the envelopes and posts them. Not caring of putting the letter in to the right envelope whatsoever.

Now the question is what is the probability that all the letters are posted incorrectly?

Though, I pity her, since the probability of getting all the letters posted correctly is only 1 in x times. What is the right value of x?

[Guest]

Bonus: if she wanted all of them to be posted incorrectly (intentional error) then she had total y ways available to her. What is the value of y?

[Guest]

All possible solutions: x = N! (N factorial) out of which only 1 right combination.

Edited February 7, 2011 by mojail

[Guest]

y = (N-1)! . (N-1) / 2

[Guest]

y = (N-1)! . (N-1) / 2

try that for N=2...

[bonanova]

One of the oddest probability questions around.

Number of ways the letters can be stuffed : N!

Number of ways the letters can be stuffed, all correctly: 1

Number of ways the letters can be stuffed, with at least 1 error: [N!] - 1

Number of ways the letters can be stuffed, all incorrectly:

We can enumerate for small values of N:

2 2 1 1/2
3 6 2 1/3
4 24 9 3/8
5 120 44 11/30

N | total | all wrong | prob

These numbers can be calculated by the formulas

Total ways = N!

Ways all wrong = (N!) x [1 - 1/1! + 1/2! - 1/3! + ... + (-1)^N/N!]

Probability all wrong: 1 - 1/1! + 1/2! - 1/3! + ... + (-1)^N/N!

The first few values for the probabilities are:

p_N = .5, .3333, .375, .3666, .36805, ...

whose reciprocals, surprisingly, are

1/p_N = 2, 3, 2.666, 2.727... 2.7169, ... which converges quickly to e.

The answers therefore are,

1. p_N(all wrong) = 1 - 1/1! + 1/2! - 1/3! + ... + (-1)^N/N!
2. x = N!
3. y = (N!) x [1 - 1/1! + 1/2! - 1/3! + ... + (-1)^N/N!]

And, if N is greater than say 10, the answers become

1. p_N(all wrong) = 1/e
2. x = N!
3. y = (N!)/e [nearest integer]

[Guest]

I've just one word for bonanova: BINGO!