Given three integers in $\{0,\ldots,100\}$ which sum up to $100$. What is the probabilty that two of them are the same?

Question

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 $0$)

Answer 1

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 $$\frac{153}{5151} = \frac{3}{101} \approx 3.0\%. $$

Answer 2

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.