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calculating odds of hitting keno

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I am occasionally dragged to a casino by other family members (and I do mean dragged; I don't like gambling much) and end up with an objective of looking for ways to lose my money slowly rather than quickly. I don't play cards and can't stand slot machines, but I have stumbled across video Keno machines, which I spend most of my time on. What I like about keno more than slots is that with slots, you are at the mercy of the games random number generator (RNG) to pick the winning/jackpot number/sequence. That is, you can play and play and play and depending on how the machine is programmed to pay out, your odds of hitting the jackpot are entirely dependent on the RNG arriving at the jackpot number. With keno however, there are probably thousands, maybe hundreds of thousands, of "jackpot" numbers/sequences picked on each spin. You just have to have 1 of those sequences in order to win the jackpot. Of course, the more you bet, the bigger the jackpot. And the more numbers that you pick each game (usually you pick anywhere from 3 to 10 numbers), the greater the jackpot.

If you're unfamiliar with keno, the game is played as follows: Typically, there are 80 numbers to pick from (1 - 80). Per game, you can pick anywhere from 3 to 10 numbers. Lets say you decide to play 6 numbers. After picking your numbers, the machine will always draw 20 numbers. If 3 of your numbers "hit", you hit for maybe 3-1 odds. 4 numbers maybe 5-1, 5 numbers around 200-1, and all 6 numbers around 1600-1.

As I said, it just "feels" that your odds are better at Keno than slots because there are jackpot numbers drawn each spin, you just have to have the right ones. With that said however, I'm at a loss as to how to calculate what those odds are. Given my keno playing experience and the fact that I have only hit once in my life (since I was only playing quarters, my "jackpot" was only $400), I suspect that the odds of hitting are pretty long. But, could one of the math experts out there determine what those odds actually are? The factors are: 80 numbers are available, I pick 6 numbers, the machine picks 20. What are the odds of my 6 numbers being among the 20 picked?

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of course a google search will find your answer explained in many different ways. here is one:

Wherever you choose to play the payout schedule overall is geared to provide around a 70% return for live Keno and around 85-90% for Online and Video Keno. This doesn't sem like a big difference but think of it in the inverse terms - the casino's profit. A game that gives you a 95% return means the casino is keeping 5% but a game that is giving you a 70% return then the casino is keeping 30% - 6 times as much! You can download my Keno Odds spreadsheet by right-clicking KenoOdds.xls and selecting "Save target as" to figure out what the casino edge is in your keno game.

The equation for calculating the probability (p) of hitting (n) numbers out of the (x) numbers you picked when (y) numbers were drawn out of (z). (i.e. - "What is the probability of hitting 4 out of the 5 numbers I picked when 20 numbers were drawn out of 80").

n = 4

x = 5

y = 20

z = 80

p(x,n) = (combin(x,n) * combin(z-x,y-x+n)) / combin(z,y)

It is important not to look at the payout schedules but to actually see how much profit the casino is keep from the money you wager. That's what the following does. I took 2 payout schedules and calculated the casino edge on all the bets and showed it in red. At first glance the 2nd payout schedule looks better because the high-end payouts are much higher but payouts are smaller on the lower end. But the payouts on the small end occur much more frequently and that's where the value of the ticket is most of the time. The 1st payout schedule shows a casino edge of about 7% while the 2nd schedule shows an edge anywhere 21-66%! You will lose your money anywhere from 3-9 times as fast playing the 2nd schedule if you choose to play the game with the big payoffs.

There are too many factors involved even amongst the video versions - so your payouts can range anywhere from 72% to 92.5% depending on where you play.

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(20/80) X (6/80) to the 6th power should give you your chances of hitting 6 out of 6.

not as simple as I thought. Tom beat me to it above.


Making the complex understandable.

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The trick is the odds of hitting a winning combo may or may not be better, but they've adjusted their paytables accordingly. You'd have to find the probability of each payout event, multiply it by the payout amount, and the totals would result in an expected value for the game. An expected value over 100% means you'd make money over time. The further under 100%, the more money you'd lose the longer you played.

Excel has a function (in the Analysis Toolpak add-on) for the hypergeometric distribution (basically that combination formula above but in easy Excel format).

(I'm doing this slightly from memory, so adjust accordingly...)

Total number of balls = 75

Number drawn each round = 20

Number picked by player = (varies but for sake of example...) 10

Number of winners "hit" by player = (varies each round but for sake of example...) 4

So, you pick 10 numbers and hit 4 of them... what's the probability? =hypgeomdist(4,10,20,75) = .16944

Pick 4 and hit 4 =hypgeomdist(4,4,20,75) = .00399

A website that I like is called "the wizard of odds". A quick google will find the website w/ that name. He's a math professor in Vegas and provides good analysis of a lot of games. Video poker, depending on the paytable, can sometimes be your best electronic game.

EDIT: hmm, I must be missing some small piece w/ that Excel formula, tried 20 numbers and 1 hit and it gave me a smaller chance than 4 numbers and 1 hit, which doesn't make sense. Oh well, you can play with it. Between the various answers, you're on the right track. (on further thought, it's the odds of "exactly" 1 right, not "more than" 1 right; w/ 20 picks it's likely to have more than 1 right, so now it makes sense to me.)

EDIT: here you go... per the tables here: http://wizardofodds.com/keno/kenoapx3.html

the probablity of getting 6 out of 6 in basic keno is 0.00013

Kurt Vonnegut: 'To be is to do'-Socrates 'To do is to be'-Jean-Paul Sartre 'Do be do be do'-Frank Sinatra

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this is where I pulled the info (I don't have an excel sheet myself)


of course, if it makes things easier, you could simply send me the $ instead. I am more than willing to turn around and send you 90% of it back, which is a better deal than most of the places.

Thanks Tom. As my clients tell me....."the checks in the mail"!!!

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