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We pick $3$ numbers (one by one) from set $\{0,1,...,100\}$. What is probabilty that two numbers are the same if sum of those $3$ numbers is $100$?

My solution: Which two are the same we can pick in $\binom {3}{2}$ ways. Suggest $x_2=x_3$- we need to find compositon $x_1+x_2+x_2=100 \implies x_1+2x_2=100$ which implies that $x_1$ is even so we can divide this by $2$. Now we get $y_1+y_2=50$ , and using formula there are $$\binom{50+2-1}{2-1}=51$$ compositions. So, probability is $$\frac{51*3}{\binom{100+3-1}{3-1}}$$

Is this right answer?

P.S.$\binom{100+3-1}{3-1}$ is number of compositions of 100 into 3 parts (allowing $0$)

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$101^3$ is certainly the wrong denominator, as it ignores the condition that the three numbers sum to $100$. –  Gerry Myerson Aug 9 '13 at 11:40
    
I think it's not wrong since sample space is just choosing 3 numbers from given set. –  Mare Aug 9 '13 at 11:55
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No, that's not the sample space, not once you add the condition about the sum being 100. Look, try the same problem with 100 replaced by 6. –  Gerry Myerson Aug 9 '13 at 11:57
    
You are right :P my bad –  Mare Aug 9 '13 at 12:02
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2 Answers

up vote 10 down vote accepted

First, we count the triples $(a,b,c)$ such that $a + b + c = 100$. For the choice of $a,b$ we need $a + b \leq 100$. Then $c = 100 - a - b$ is uniquely determined. For $a + b \leq 100$, the number of possibilities is $$101 + 100 + 99 + \ldots + 2 + 1 = \frac{102\cdot 101}{2} = 5151.$$

Now we count the triples with the additional condition that two numbers are the same. Note that $a = b = c$ is not possible (otherwise, $100 = a + b + c = 3a$ would be a multiple of $3$). So there are $3$ possibilities to choose the identical pair of numbers. For a single such choice, say $a = b$, we have the $51$ possibilities $a\in\{0,\ldots,50\}$ s.t. $a + b = 2a \leq 100$. Again, $c$ is uniquely determined. So in total, there are $3 \cdot 51 = 153$ triples $(a,b,c)$ such that two numbers are the same and $a + b + c = 100$.

This gives the resulting probability of $$\frac{153}{5151} = \frac{3}{101} \approx 3.0\%. $$

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"For a+b≤100, the number of possibilities is 100+99+98+…+2+1=50⋅101=5050". That should be 5151 rather than 5050 as a and b can take values zero? –  Johannes Aug 9 '13 at 13:04
    
@Johannes. Right, it's corrected now. Thanks! –  azimut Aug 9 '13 at 13:06
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Isn't 3*51 153? (FWIW, a brute force calculation I just did gave 153/5151.) –  DSM Aug 9 '13 at 13:10
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$153/5151=3/101$. If the final form of the answer is that simple, there must be a simpler way of getting it. –  Gerry Myerson Aug 9 '13 at 13:11
    
@DSM: Well, of course it is. Thanks! –  azimut Aug 9 '13 at 13:13
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Picking three numbers from the set $\{0,1,2,...,6N-2\}$. What is probability that two numbers are the same given the sum of those three numbers is $6N-2$?

Firstly, the number of triplets adding up to $6N-2$ is given by the triangular number $3N(6N-1)$. Out of these, the $3N$ triplets $(0,0,6N-2), (1,1,6N-4), .. , (3N-1,3N-1,0)$ and their permutations contain two equal numbers. So we have $9N$ triplets out of $3N(6N-1)$ triplets that have two identical numbers. The probability sought is $3/(6N-1)$. For N=17 we recover OP's question.

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