FILE No.40Unsolved
In a coffee shop, three customers (A, B and C) each left one identical-looking umbrella in the stand. When they left, the three took one umbrella each, completely at random, without telling them apart.
Later it turned out that "all three people went home with an umbrella that was not their own." The shop owner was surprised: "Such a coincidence hardly ever happens."
When three people take umbrellas at random, what is the probability that everyone gets someone else’s?
If you get stuck, open the hints one at a time, in order.
In how many ways can the three umbrellas be arranged in all?
In how many arrangements does no one take his own umbrella?
Probability = the number that meet the condition / the total number
When three people take umbrellas at random, what is the probability that everyone gets someone else’s?
Choose the answer closest to the truth.
1/3
The answer is 1/3.
There are 3 x 2 x 1 = 6 ways for the three to take the three umbrellas. Of these, only two leave everyone with an umbrella other than their own: "A takes B’s, B takes C’s, C takes A’s" and "A takes C’s, B takes A’s, C takes B’s."
The probability is 2 / 6 = 1/3. That is about once in every three times, so it is not a "rare coincidence."
The feeling that "everyone getting it wrong is rare" makes you estimate the probability far lower than it really is.
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