EuroMillions is a game where you have to tick 5 numbers in a grid of 50 numbers (from 1 to 50) (there is also a stars system, but let's omit it). Then, the draw: 50 balls numbered from 1 to 50 in box, 5 get picked up (so there are 45 balls left in the box). If the 5 numbers you ticked equal the 5 numbers of the balls that have been picked up: jackpot!
I'm looking for the probability that one given number - let's say the number 3 - gets picked up in a draw.
I was thinking about the following formulae:
1) 1/50 * 5
// wrong I guess, because every time you pick up a ball,
// there are n-1 balls left in the box
2) 1/50 + 1/49 + 1/48 + 1/47 + 1/46
// it must be something like that,
// but it would be great if somebody could confirm ;)
Could any math brain help me?
The number of options to pick 5 numbers: $50\choose5$
The number of options to choose the number 3, then 4 other balls: $49\choose4$
Total probability: $p(A)=\frac{|A|}{|\Omega|}=\frac{49\choose4}{50\choose5}=\dfrac{\frac{49! }{45!4!}}{\frac{50!}{45!5!}}=\frac{49!45!5!}{50!45!4!}=\frac{5}{50}=0.1$
Your second attempt is not true, since it need to be $\frac1{50}$ for the first, $\frac{49}{50}*\frac1{49}$ for the second, since the second is three only if the first is not three, etc.