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February 18th, 2019, 10:22 AM   #1
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Labeling on bottles probability

Hi, I'm here to check if my answer to a question is right:

Anne asks her friend Sharon to label 4 bottles of perfume in her room, giving her a list which labels should be added to which bottle. Sharon loses the list and labels the bottles at random. When Anne returns she notices that each bottle contains the wrong label. What is the probability of this having happened?

I named the bottles A,B,C and D.
For each bottle to be matched with the wrong label the probability would be:
A - 3/4
B - 2/3
C - 1/2
D - 1/1

3/4 x 2/3 x 1/2 x 1/1 = 1/4

I got 1/4. Is this right and is there a different way of solving this question?

Last edited by Kieran1; February 18th, 2019 at 10:29 AM.
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February 18th, 2019, 01:26 PM   #2
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It is more complicated. Let A choose label B, B choose label C, and C chooses label A, then D is stuck with label D. In general for A the probability is 3/4, but for B the probability depends on whether or not A chose B, resulting in a probability 1 for B, etc.
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