Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

We randomly put $100$ numbered balls in $100$ baskets, and if I am to ask what's the probability of the third ball being in a basket between the first and second ball, I know the answer is exactly one third, intuitively. A similar case would be asking what's the probability of one of these balls being bigger than the other. The difference is that for the latter example (with the probability being $1/2$), I can formally prove it using the law of total probability. In the first example I was not able to devise or think of a simple proof, so I was wondering if you could perhaps show me how it can be done using formality rather than common sense.

Thanks a million!

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Assuming only one ball can go in each basket, you can ignore all the other balls and look at the order of balls $1,2,3$. There are $3!=6$ orders of these balls, and ball $3$ is in the middle in $2$ of them.

share|improve this answer
    
I saw that right after typing, removed comment - but thanks for following up! –  gnometorule Feb 20 '13 at 4:54
    
What's the justification for ignoring the other balls? I was able to get to this situation but didn't know how to justify it –  NBP Feb 20 '13 at 5:12
2  
@NBP The justification is, basically, that in the question you're asking, all of the other 97 balls are meaningless. The question you're asking is "what is the probability that ball 3 is between ball 1 and ball 2?" It doesn't matter if ball 1 is in the first basket, and ball 2 is in the 93rd basket. Imagine you line up all these baskets on a cliff. Shove all the baskets containing balls numbered 4 to 100 off the cliff. You're left with balls 1, 2 and 3. There are only 6 ways these can be arranged, regardless of their original position in the line of 100 baskets. –  Arkamis Feb 20 '13 at 18:42

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.