In a new lotto game, you must choose six numbers from the numbers 1 to 25. Six winning numbers are drawn, th
May 11th, 2009 | by admin |
Weekly Reader Insider s asked:
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 is the six numbers you have chosen contain exactly three of the winning numbers and the supplemental number. Find the probability of winning Fifth Prize.
Todd
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 is the six numbers you have chosen contain exactly three of the winning numbers and the supplemental number. Find the probability of winning Fifth Prize.
Todd
One Response to “In a new lotto game, you must choose six numbers from the numbers 1 to 25. Six winning numbers are drawn, th”
By MathMan TG on May 11, 2009 | Reply
Let’s call the the 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%