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### Probability of Matching Envelopes and Letters

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Date: 03/03/98 at 04:20:34
From: Raveen Dey
Subject: Probability- permuatations

Six letters were to be placed in envelopes for posting. Unfortunately,
the letters were dropped before being put in their envelopes, and they
were placed in at random.

a) What is the probability that none is in the correct envelope?
b) What is the probability that exactly one is in the correct
envelope?

Answers
a)53/144
b)11/30
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Date: 03/03/98 at 08:14:40
From: Doctor Anthony
Subject: Re: Probability- permutations

I answered a nearly identical question earlier today, so I will copy
that answer here. I was considering 5 letters and 5 envelopes.

If we let A be the event that letter A is in ist correct envelope and
similarly B is the event that letter B is in its correct envelope,
then

P(A) = 1/5 and

P(A and B) = 1/5 * 1/4

Now use the inclusion-exclusion principle to get probability that A or
B or C .... or E are correctly placed.

P(A or B or C .... or E) = P(A) + P(B) + P(C) + P(D) + P(E)
- P(A and B) - P(B and C) -....
+ P(A and B and C) + P(B and C and D) + ....
- P(A and B and C and D) - P(...) -......
+ P(A and B and C and D and E)

= 5*(1/5)
- 5C2*(1/5)(1/4)
+ 5C3*(1/5)(1/4)(1/3)
- 5C4*(1/5)(1/4)(1/3)(1/2)
+ (1/5)(1/4)(1/3)(1/2)(1/1)

5*4    1    5*4*3     1      5*4*3*2       1          1
= 1 - --- * --- + ----- * ----- - --------- * ------- + ---------
1*2   5*4   1*2*3   5*4*3    1*2*3*4    5*4*3*2   5*4*3*2*1

= 1 - 1/2! + 1/3! - 1/4! + 1/5!

This is the probability that at least one letter is correctly placed.

The chance that none is correctly placed is 1 - (above result), or

= 1 - 1 + 1/2! - 1/3! + 1/4! - 1/5!

= 1/2! - 1/3! + 1/4! - 1/5!

With n letters and envelopes, the probability that none is correctly
placed is:

= 1/2! - 1/3! + 1/4! - 1/5! + ....... + (-1)^n 1/n!

Note that as n becomes very large, this probability tends to the value
e^(-1), since:

e^(-1) = 1 - 1 + 1/2! - 1/3! + 1/4! - ........ to infinity

= 0.367879...

With six letters and envelopes, the probability that none is correctly
placed is:

= 1/2! - 1/3! + 1/4! - 1/5! + 1/6!

= 1/2 - 1/6 + 1/24 - 1/120 + 1/720

= 53/144.

To find the probability that exactly one is correctly placed, we need
to find the total number of ways with 1 correct and 5 incorrect.

We can use the result of the last problem to see the number of ways
where 5 letters are all incorrect.

If we had 5 letters and 5 envelopes, the probability that all are
incorrect is 1/2! - 1/3! + 1/4! - 1/5!

Therefore the number of arrangements with all incorrect is given by

5!(1/2! - 1/3! + 1/4! - 1/5!) = 44

Now imagine the letters laid out in a row, and the envelopes in a
matching row.

If the first letter is the only correct one opposite its envelope,
then there are 44 arrangements for the other 5 all putting them in the
incorrect envelope. Similarly, if the second one is the only correct
one, then this can be associated with 44 arrangements for the other 5,
which will all be incorrect. Similarly, for third, fourth, fifth and
sixth letters, they can each be associated with 44 arrangements,
making the other 5 all incorrect.

So the total number of arrangements in which just one is correctly
placed is:

6 x 44 = 264

The total number of possible arrangements is of course 6! = 720, so
the probability that exactly one is correct is

264/720 =  11/30

-Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Probability

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