# [Math] How to Calc the odds of winning a lucky dip

probability

How do I calculate the odds of winning?
I am doing a lucky dip raffle – you pay $£1$ and pick out $3$ balls, there are $495$ balls and $50$ prizes. Each ball has a number on, and if the number matches one of the $50$ prizes you win. What are the odds of winning – bearing in mind each turn you take you pick out $3$ balls?

We will calculate the probability of winning at least one prize if we draw $3$ balls.
It is easier to first calculate the probability of winning no prize. Imagine drawing the balls one at a time, and looking at them. The probability the first ball is a loser is $\frac{445}{495}$.
Given that the first ball was a loser, the probability the second ball is a loser is $\frac{444}{494}$. So the probability the first two balls are losers is $\frac{445}{495}\cdot\frac{444}{494}$. Given that the first two balls were losers, the probability the third one is a loser is $\frac{443}{493}$. So the probability of $3$ losers in a row is $$\frac{445}{495}\cdot\frac{444}{494}\cdot{443}{493}\approx 0.726051856.$$ The probability of at least $1$ winner is therefore approximately $1$ minus the above number. This is about $0.273948$.
If you want to put this in the language of odds, the odds of winning at least one prize are about $0.273948$ to $0.7260518$, roughly $0.377312$ to $1$.