# Computing the probability of selecting two numbers without replacement from the set $\{1, 2, 3, \ldots, 157\}$ such that both are divisible by $3$?

I have a set of numbers $\{1,2,3,...,157\}$.

How to compute the probability of selecting two numbers without replacement such that are both divisible by $3$?

• Are we selecting $2$ numbers, without replacement? – André Nicolas Oct 24 '15 at 9:22
• @AndréNicolas, with replacement you mean that we can 'repeat' the number? Then no - we have the set and we can't repeat the number. – Ganjira Oct 24 '15 at 9:26
• There are $52$ multiples of $6$ in our set, $3\cdot 1$ to $4\cdot 52$. Imagine picking the numbers one at a time. The probability the first is a multiple of $3$ is $\frac{52}{157}$. Given the first is divisible by $3$, the probability the second is divisible by $3$ is $\frac{51}{156}$. So our required probablity is $\frac{52}{157}\cdot\frac{51}{156}$. This can be simplified. – André Nicolas Oct 24 '15 at 9:33
• By selecting "the" numbers, do you mean any two numbers ? – true blue anil Oct 24 '15 at 9:33
• Any 2 numbers from the set of numbers I mentioned above. – Ganjira Oct 24 '15 at 9:44

Multiples of $3$ in the given range are $3\times1$ through $3\times52$, i.e. numbering $52$
Thus $Pr =$ $\dbinom{52}{2} \over \dbinom{157}{2}$