LOTTERY GAME
In a new lotto game, you must choose six numbers from the numbers 1 to 25. Six winning numbers are drawn, then one supplemental number is drawn from the remaining numbers. You win Fifth Prize if the six numbers you have chosen contain exactly three of the winning numbers and the supplemental number. Find the probability of winning Fifth Prize.
The answer is currently being disputed.
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1 solution
Let's call the set of 6 winning numbers the WS and the supplemental number S.
What are the chances that OUR six numbers have three of the WS and S ?
Ways to have 3 of the WS: 19 x 18 x 17 x 6 x 5 x 4 / 6! (19 of the numbers are not WS, after choosing one of them there are 18, then 17. Then we pick WS numbers, with 6, then 5, then 4 choices. They can be in any order, so we divide by 6!)
But to win 5th place, one of the first three must be S. So we switch to 1 x 18 x 17 x 6 x 5 x 4 / 6! out of 25 x 24 x 23 x 22 x 21 x 20 / 6! (total ways to pick 6 numbers) The 6!s cancel, so it's 18 x 17 x 6 x 5 x 4 out of 25 x 24 x 23 x 22 x 21 x 20 or 36720 / 127512000 or 51 / 177100 including the 6!s 0.00028797... that is, less than 3 in 10000 or 0.03%