Question

Suppose that four guests check their hats when they arrive at a restaurant, and that these hats are returned to them in a random order when they leave. Determine the probability that no guest will receive the proper hat.

Answer 1

@satellite73

Answer 2

i think we can do this

Answer 3

the probability that the first one does not get her hat is \(\frac{3}{4}\)

Answer 4

leaving 3 hats, so the probability that the second guest does not get her hat is \(\frac{2}{3}\)

Answer 5

hmm maybe that is not right, hold the phone

Answer 6

definitely wrong, sorry lets try again

Answer 7

mb we should try with 3 persons to ease?

Answer 8

to find probability of 3 people not getting their hat instead of 4

Answer 9

this is called a "derangement"
there is a formula for it

Answer 10

or if you don't want to use the formula, count
there are \(4!\) ways to distribute the 4 hats
of those i believe there are 9 ways that they do not match
\(4!=24\) so it is not too many to list

Answer 11

i.e. start with ABCD as the correct order, then start listing
ABCD (4 matches)
ABDC (two matches)
ACBD (2 match)
ACDB (1 match)
ADBC (1 match)
ADCB (2 match)

obviously you have to start with something other than A

Answer 12

if i am not mistaken you will count 9 and get \(\frac{9}{24}\) but we can also try the formula
\[\sum_{k=0}^4\frac{(-1)^k}{k!}\] and see if we get the same answer

Answer 13

wow how to you like that!
http://www.wolframalpha.com/input/?i=sum+k+%3D0+to+4+%28-1%29^k%2Fk!

Answer 14

wait a minute i'll try to count it manually

Answer 15

i got 7/24

Answer 16

thank you!