How to solve Odds for a lottery with 2 result sets ( not a conventional type )

Question

Hoosier lottery has a game Quick Draw with bullseye. The game consists of 20 balls drawn 1-80 with the bullseye being randomly drawn after from one of the 20 numbers. The player to win picks 10 numbers 1-80 with matching all 10 to the 20 picked and the Bullseye as one of your numbers for the top prize.

I have looking at this as a 2 part problem basically solve as a regular binomial equation for the games draw (n:k) = (n!/(k! * (n-k)!) . the posted odds for the game are : 1 in 8,911,711 match 10 of 10 and 1 in 17,823,422 match 10 of 10 and the bullseye. the actual numbers I get starting this : ( big numbers incoming )

n=80 k= 20 the user picks are 10

n!= 71,569,457,046,263,802,294,811,533,723,186,532,165,584,657,342,365,752,577,109,445,058,227,039,255,480,148,842,668,944,867,280,814,080,000,000,000,000,000,000

k! = 2,432,902,008,176,640,000

(n;k)= 3,535,316,142,212,174,320

This is where I am stuck, nothing i have tried gets me close to the posted Odds. My thoughts are now that i have a result set I could use them as n! or k! but again it is not working out . Or should I be looking at some fractional part of the problem since we are introducing basically a 3rd factor in this ?

So the actual question is how do I solve for the odds in this game and am I approaching it correctly and if not what approach should I be looking at ?

this seems like the old candies in a jar example with a slight twist in that you are going to select some random candies first then you have to match them after the fact but compute the odds on the full jar ...I am just NOT getting my head wrapped around it ..

Answer 1

In calculating the odds, (80:20), the number of ways the lottery can select its 20 numbers,

never enters into the picture.

Think of it this way:

"chances of winning"

= "# ways to choose winning numbers" / "# ways to choose any numbers"

= "# ways lottery choses 20 numbers" × "# ways user choses 10 numbers to match the chosen 20"

/ "# ways lottery choses 20 numbers" × "# ways user choses 10 numbers"

Therefore "# ways lottery choses 20 numbers" cancels out in the fraction.

So you can calculate the odds by assuming having fixed chosen 20 numbers plus a BullsEye number.

First without BullsEye:

"chances of winning"

= "# ways user choses 10 number to match the chosen 20" / "# ways user choses 10 number"

= (20:10) / (80:10) = 1/8,911,711

Now with the BullsEye:

"chances of winning"

= "# ways user choses the BullsEye number" × "# ways user choses 9 numbers to match the remaining 19"

/ "# ways user choses 10 numbers"

= 1 × (19:9) / (80:10) = 1/17,823,422