Please confirm or correct keno question probability

The game is Keno. The player picks 3 numbers, and gets 3 consecutive draws to hit each of those 3 numbers at least once. I was giving 4-1 odds (a little 8.25% juice for me) on this to players under the assumption that the chances of this were 18.35% of winning. Is 18.35% the correct probability of hitting 3 separate numbers at least once each in 3 draws? He could miss all of them the first 2 draws and still win if he hit all 3 on the last draw, so just to clarify the language, he needs to hit each number at least once, but he doesn't have to hit one on every draw to win.

Personally, I figured there were 554,601,821,696,000 combinations with 101,795,901,960,000 of those hitting all 3 numbers, but I'd love to make sure I figured correctly.

Quote: shakhtar

The game is Keno. The player picks 3 numbers, and gets 3 consecutive draws to hit each of those 3 numbers at least once. I was giving 4-1 odds (a little 8.25% juice for me) on this to players under the assumption that the chances of this were 18.35% of winning. Is 18.35% the correct probability of hitting 3 separate numbers at least once each in 3 draws? He could miss all of them the first 2 draws and still win if he hit all 3 on the last draw, so just to clarify the language, he needs to hit each number at least once, but he doesn't have to hit one on every draw to win.

Personally, I figured there were 554,601,821,696,000 combinations with 101,795,901,960,000 of those hitting all 3 numbers, but I'd love to make sure I figured correctly.

How did I miss this?
I get 0.189326874129085 which is off by yours by exactly the number of ways to pick 0 right followed my all three.

Quote: miplet

How did I miss this?
I get 0.189326874129085 which is off by yours by exactly the number of ways to pick 0 right followed my all three.

Great job. I rechecked my original work and found the exact omittance you referred to. The sequence of 0 hits on the first draw (34,220) - 3 hits on the second draw (1140) - followed by all on the 3rd draw (82,160).

34,220 x 1140 x 82,160 = 3,205,127,328,000 which when added to my earlier incorrect total of 101,795,901,960,000 will give us the now correct total of 105,001,029,288,000 which is exactly .1893268741290

I even decided to fire a rapid fire simulation of 100 games and eerily got 19 successes.

So I was only taking around 5.35% juice on the bet. A little low when giving 4-1 on low volume, but still worth it to give action, since it's a cool action bar game with keno draws every 4 minutes.