This is my first thread on this board, so I hope I have the correct forum, and my question is not inappropriate (sp?).
Everyone probably is familar with Keno. My wife and I have an ongoing argument about the odds I hope someone can answer.
Say out of 80 numbers you can pick a max of 10. The game draws 20 at random. If you pick 10 numbers, you must hit 4 out of 10 for a minumum win. If you pick only 7 numbers, you only need 3 hits for a minimum win.
I say that picking 7 and hitting 3 out of 7 is better odds than picking 10 and hitting 4 out of 10.
Remember, 80 numbers, 20 drawn. Whos is correct? My wife or myself ?
Thanks, bubinski.
Without doing all the higher math, I would say whichever bet pays more has the least likely chance of winning.
You just said that you’re gambling, if there was a science to it then it wouldn’t be a game of chance. Slots have always been the “sucker’s bet” in Vegas, but last time I went there the only advice I ever got was “don’t play Keno, ever, It’s a sucker’s bet”. This is certainly not meant to cast aspersions on your game of choice, but I have to agree with the fact that if there’s NO playing involved, then you must be at the house’s mercy…otherwise that game would be banned from their casino.
Firstly, welcome to the boards, bubinski.
Secondly, Keno is certainly not familiar to everyone on these boards, but Google should be.
Googling for keno odds, the first hit gives the probabilities as:
P(3 from 7) = 0.1750
P(4 from 10) = 0.1473
so, I guess you were right.
Why would you bet on any game where the house knows your numbers going in, but you don’t know their numbers until they are drawn?
:confused
AFAIK, no one has ever hit the jackpot in the history of Keno. Supposedly, you’re more likely to die by rolling out of bed and breaking your neck some 20 times in a row than to hit the jackpot in Keno. How you die 20 times in a row is beyond me, but that’s what I’m told 
Once upon a time, when I was a grad assistant teaching freshman classes, I allowed them to figure the expected return on the keno games in the local bars as an extra credit assignment (I believe I made the payoff schedule for one popular watering hole available to students who did not want to conduct their research first hand). I was rather surprised at just how bad they turned out to be - the best choice card (I don’t remember how many spots) returned about $0.50 on the dollar. Some of them returned about $0.20. Gambling games do not need to be that terrible to make good money for the business running the game. They could have returned about $0.80 on the dollar and done just fine. But then again, people were fool enough to play the game with the payoff schedules as they were.
Background - it’s an exercise in computing hypergeometrics, which is the reason I used it as an assignment. The obvious way to think about it is that the probability of drawing W winners from choosing S spots on the card when K numbers are drawn out of N is:
C(K,W)*C(N-K,S-W)/C(N,S)
Where C(n,k) is the “combinations of k things chosen from n” = n!/(k!*(n-k)!).
N = 80, K = 20, S = 7 or 10, and W = 3 or 4 in the OP’s original problem. You’ll get the numbers given in TGU’s link.
If they thought about what they were calculating, they could reduce their effort a good bit - for instance, treating the spots as the “distinguished” objects rather than the drawn numbers leads to a constant denominator (C(N,K)) for all of the cards. That actually occurred to a couple of them. That they could iteratively compute the P(0), P(1), P(2) answers by multiplying the previous result by a fraction computed from 4 values occurred to none of them.
[Hijack]
I have heard blackjack has the best odds of any casino game. But after watching people hit an 18 when the dealer holds 16 (has to hit), I have to conclude: It’s not so much the odds of winning as how you play the game.
[/Hijack]
I was just in Vegas a couple of weeks ago and watched a guy split 2 10s against the dealers 19.
I got up and left after that…