SOLUTION: An absent-minded secretary randomly puts four letters into four envelopes. What is the probability that at least one of the letters will be in the correct envelopes.

Question 1202976: An absent-minded secretary randomly puts four letters into four envelopes. What is the probability that at least one of the letters will be in the correct envelopes.
Found 2 solutions by Peter Bai , greenestamps:
Answer by Peter Bai (1) (Show Source): You can put this solution on YOUR website!
Answer=5/8
At least 1 envelope
Therefore
Minimum (1) to maximum (4)
Number of ways in which the envelopes can be arranged=4!
Now
P=4C1/4!+4C2/4!+4C3/4!+4C4/4!
P= 5/8
Answer by greenestamps(13198) (Show Source): You can put this solution on YOUR website!

The response from the other tutor arrives at the correct answer only by chance; the method used is faulty. If we use that method on the problem with only 3 envelopes, the answer we get is

3C1/3!+3C2/3!+3C3/3! = 3/6 + 3/6 + 1/6 = 7/6

which of course is not a valid number for a probability.

The probability that at least one letter gets into the right envelope is 1, minus the probability that none of them get in the right envelope.

If no letter gets in the right envelope, we have a derangement -- an arrangement in which none of the letters are in the right place. The number of derangements of 4 items is

(N.B. -- Do an internet search on "derangement" if you want to try to understand that formula....)

Since there are 4! = 24 possible arrangements of the 4 letters, the probability that we have a derangement (none of the letters is in the right envelope) is 9/24 = 3/8

So the probability that at least one letter gets in the right envelope is 1-3/8 = 5/8.

ANSWER: 5/8

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