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Almost Winning the Lottery

Almost Winning the Lottery


in the ‘winning the lottery’ section of probability, i understand everything explained, but often in lotteries, you can win a prize if you get any combination of 2/3/4/5 out of 6 balls etc.
to deepen my understanding, I wanted to work out, what is the probability of drawing a combination of any 2 of 59 balls when 6 balls are the total selected (same as English lotto) – 
according to the lotto website the odds of drawing 2 balls is 10.3, https://www.lottery.co.uk/lotto/odds
i have tried to work this out myself but struggling, i have spent 2.5 hours trying to work it out on all the teaching to this point, can anyone help please?
The odds of wininng are 59!/(6!*53!), the odds of any one ball appearing – 1 in 59, or  59!/(1!*59!) but the odds of any 2 balls being drawn is not as I understand it:

  • (1×59)*(1×58) = 3422 this would give you the permutations of 2 balls from 59 being drawn and if only 2 balls could be drawn
  • 59!/(2!*57!) = 1711 as this would give you the chances of winning the lottery if 2 balls were all that could be drawn, if 2 balls were the maximum, but in the lottery after ball 1 is drawn there are 5 more balls, any of which could be a match, how do we account for this?
  • 59!/(2!*57!) / 58!/(5!*53!) = 2678
  • 59!/(6!*53!) / 58!/(5!*53!) = 9.83

the last calculation being the closest but still not correct???
I just cannot work it out…..
Thanks in advance.

1 Answer

365 Team

Hey Daniel, 
So, we’re assuming we pick 6 numbers out of 59 and we want to know the odds of having at least 2 correct. 
The total number of outcomes would be 59! / 53! 6! = 45,057,474. Now, if we assume that we’re getting 2 of the numbers right, that means we have 6!/2!4! = 15 ways of picking the two we get correct.
Then, out of the 53 numbers we didn’t select, we need to get exactly 4. That can be done in 53!/4!49! = 292,825 – many ways. That means that for any combinations of two out of our 6 numbers, we have 292, 825 different ways of winning. That means, that in 15 * 292,825 = 4,392,375 of all cases, we have exactly 2 numbers correct. 
Then, when we divide the number of total outcomes (45,057,474) over the number of winning outcomes (4,392,375), we get the odds of having exactly 2 numbers right. Hence, the odds we’re looking for are 45,057,474 / 4,392,375 = 10,25811184, or approximately 10.3.
365 Vik 

thanks for the explanation, appreciate that. i have just studied your answer for an hour trying to work it all out in Excel and now it makes sense, i have also then worked out from this how to work out your overall odds of winning any given prize, all very interesting. thanks again

9 months