SOLUTION: 50 students have graduated from a univeristy. As, they pose for a picture they throw their hats in the air. After the caps have been thrown and the picture taken, all of them rando

Algebra ->  Probability-and-statistics -> SOLUTION: 50 students have graduated from a univeristy. As, they pose for a picture they throw their hats in the air. After the caps have been thrown and the picture taken, all of them rando      Log On


   

Question 658660: 50 students have graduated from a univeristy. As, they pose for a picture they throw their hats in the air. After the caps have been thrown and the picture taken, all of them randomly select a cap.
a) What is the probability that at least one of them retrieves his/her own cap?
b) What is the probability that none of them retrieves his/her own cap?

My attempt:
I tried to use the standard for for summation which looked like this after i plugged in the values:
n=50
e^50=1+50+50^2/50! +50^3/30! .......50^50/50!
but solvine this would take ages
is there a shortcut to solving this problem?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Hint:

Parts a) and b) are mutually exclusive and complements of each other. In other words,

P(At least one gets their own cap) + P(No one gets their own cap back) = 1

So if you can find one of the probabilities above, then you can automatically find the other.

It's easiest to find P(No one gets their own cap back) and you can use a derangement to do so.

A derangement is a permutation in which something like ABCDE rearranges to EDABC. This new arrangement is where all letters have moved (no single letter is in the same spot).