How likely is it to guess three numbers? In the Irish lottery if you guess three numbers correctly you win 576x your original stake and there are 12 draws a week.
My questions is: How likely is it, over the course of two years (104 weeks or 1248 draws) that I will guess the numbers correctly?
The rules are as follows:


*

*you can only choose numbers between 1-49

*you can only choose three of these numbers

*the total draw will produce 6 numbers, yet only the three you have chosen need to match


So over two years, how likely will I be to win? Thanks :)
 A: At a given draw you are choosing one of ${49\choose3}=18424$ triples at random. Since $6$ numbers will be drawn there are ${6\choose3}=20$ successful triples. It follows that the probability $p$ of a success in one draw is given by
$$p={20\over18424}\doteq0.00108554\ .$$
The probability $q$ that you fail in all of 1248 draws is therefore given by
$$q=(1-p)^{1248}\doteq0.25782\ .$$
Therefore you can count on succeeding at least once in two years with probability $1-q\doteq0.74218$.
A: The chance of getting the first number right is $\frac6{49}$, the second one is $\frac5{49}$ and third one is $\frac4{49}$. the product of these 3 fractions gives you the probability of winning a draw on a single attempt. If you have $1248$ draws, multiply this fraction by the same to get the probability of winning $1$ (or more) draws over $2$ years.
Note:
If numbers can not be repeated, use $49$, $48$ and $47$ as denominators, instead of $49$ all $3$ times.
Assuming numbers can be repeated, this gives a result of about $1.2$, which means you can be pretty sure of winning once in $2$ years. $576$ times profit is still an inadequate amount for $1248$ tries, but its worth trying your luck.
