## A Keno Game Problem

It’s been a while since I posted anything in the Cute Problems category so, given that I’ve got an unexpected gap of half an hour today, I thought I’d return to one of my side interests, the mathematics and games and gambling.

There is a variety of gambling games called *Keno* games in which a player selects (or is given) a set of numbers, some or all of which the player hopes to match with numbers drawn without replacement from a larger set of numbers. A common example of this type of game is Bingo. These games mostly originate in the 19th Century when travelling carnivals and funfairs often involved booths in which customers could gamble in various ways; similar things happen today, though perhaps with more sophisticated games.

In modern Casino Keno (sometimes called Race Horse Keno) a player receives a card with the numbers from 1 to 80 marked on it. He or she then marks a selection between 1 and 15 numbers and indicates the amount of a proposed bet; if n numbers are marked then the game is called `n-spot Keno’. Obviously, in 1-spot Keno, only one number is marked. Twenty numbers are then drawn without replacement from a set comprising the integers 1 to 80, using some form of randomizing device. If an appropriate proportion of the marked numbers are in fact drawn the player gets a payoff calculated by the House. Below you can see the usual payoffs for 10-spot Keno:

If fewer than five of your numbers are drawn, you lose your £1 stake. The expected gain on a £1 bet can be calculated by working out the probability of each of the outcomes listed above multiplied by the corresponding payoff, adding these together and then subtracting the probability of losing your stake (which corresponds to a gain of -£1). If this overall expected gain is negative (which it will be for any competently run casino) then the expected loss is called the house edge. In other words, if you can expect to lose £X on a £1 bet then X is the house edge.

What is the house edge for 10-spot Keno?

Answers through the comments box please!

Follow @telescoper
July 25, 2014 at 4:50 pm

Found it to be 0.9965280074£

July 25, 2014 at 5:01 pm

Nope!

However your answer did make me check the numbers in the Table and I found an error: the payoff for 6 numbers should have been £17; I’ve now changed it. This may make a significant difference, so try your calculation again.

July 25, 2014 at 5:11 pm

Makes it about 0.99535856

July 25, 2014 at 5:11 pm

Nope, that’s way too high….

July 30, 2014 at 11:40 am

“Twenty numbers are then drawn without replacement from some form of randomizing device.”

I take it that this opaque urn contains each number from 1 to 80 once?

July 30, 2014 at 11:47 am

Yes, that’s right. Usually it’s like one of those machines used in Bingo or the lottery…

September 19, 2014 at 3:46 pm

Approximately -0.207£ for the expected payoff, which gives a house edge of 0.207£, if I’ve interpreted edge correctly.

September 19, 2014 at 4:14 pm

That’s right. The house edge in this game is about 21%.