Electronic Bingo Pattern Priority

Hello all! I'm trying to figure out the probability of certain bingo patterns. To simplify the problem, let's assume a single bingo card, 75 total balls, and we will draw 35 balls.

I'm clear on how to calculate the probability of any particular pattern consisting of k hits within N ball draws: multiply the probability of catching k hits in N balls drawn with the probability of forming said pattern on a bingo card given k hits summed over the appropriate values for k. However, many machines stipulate that in the case of multiple patterns being hit, the one with higher (sometimes lower) priority will take precedence. For example, if a machine pays out for a coverall in 26 draws, and in addition, pays out for 4-corners in 30 draws, getting the coverall win would count as getting both. The machine would then choose one of the two prizes based on some rule. For our example, lets say the coverall has a higher payout and will get chosen over the 4-corners.

My question is, how can we calculate the RTP/probability of getting one of the 4-corners in 30 draws when getting the coverall in 26 draws effectively "blocks" some of the times the 4-corner pays out. Thanks!

I would just subtract the coverall probability in 26 draws from the four corner probability in 30 draws.

Not that you asked, but the math can get very messy when lots of patterns get involved. When I need to design a class 2 game, I do it by simulation.

Thanks Wizard!

Well you saw where this was going; I'm trying to solve a simple two pattern case in order to understand how to scale to many patterns possibly "blocking". I'm a sucker for closed-formed/mathematical solutions. When you say simulation, do you mean Monte Carlo? Or are you iterating through all possible outcomes to come to an exhaustive solution?

Here's how I do it, and the number matches what I get in simulation:

There are C(71,26) combinations of the first 30 numbers drawn that include all four corner numbers.
There are C(71,26) * 30! permutations of these numbers. (Permutations have to be used because the order gets to be important.)

There are C(51,2) combinations of the first 26 numbers drawn that include all 24 numbers on the card.
There are C(51,2) * 26! * P(49,4) permutations of the first 30 numbers.
("Where do you get that from?" There are C(51,2) combinations of the first 26 that have all 24 numbers; each combination can be in any of 26! orders; each of the remaining permutations of 4 of the 51 undrawn numbers is equally likely to be the 27th through 30th numbers drawn.)
Of each of these permutations, there is a 1/6 chance that the last of the 24 numbers on the card drawn is a corner number, which means the four corners were not drawn first.
Thus, there are a total of C(51,2) * 26! * P(49,4) / 6 permutations of the first 30 numbers where not only are all 24 numbers on the card in the first 26 drawn (this is why order is important), but the last number drawn was one of the four corner numbers, so the four corners did not come first.
Subtract C(51,2) * 26! * P(49,4) / 6 from C(71,26) * 30! to get the number of "winning" permutations in the first 30.
Divide by P(75,30), which is the total number of permutations of 30 numbers from the 75, and you get about 1 / 44.35.

Quote: Wizard

I would just subtract the coverall probability in 26 draws from the four corner probability in 30 draws.

The problem with this is, you are also subtracting the instances where the four corners are drawn before the other 20 numbers on the card.

Quote: ThatDonGuy

The problem with this is, you are also subtracting the instances where the four corners are drawn before the other 20 numbers on the card.
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Maybe I misunderstand the rules. In the case of overlapping patterns, does the player win the higher of the two payouts?

Quote: Wizard

Maybe I misunderstand the rules. In the case of overlapping patterns, does the player win the higher of the two payouts?
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In most two part bingo games, it’s hard if not impossible to achieve both part one and part two at the same time.

However sometimes it can be done. For example if part one is a regular bingo, and part two is a postage stamp, either winning combination can occur first and on rare occasions it can occur at the same time, and even on the same board.

Yes, in that case, the player would be entitled to both prizes.

Quote: DJTeddyBear

However sometimes it can be done. For example if part one is a regular bingo, and part two is a postage stamp, either winning combination can occur first and on rare occasions it can occur at the same time, and even on the same board. link to original post

A single bingo and postage stamp do not completely overlap each other, thus I would expect the player to get paid for both.

I still don't understand what is wrong with my initial answer that Don challenged.

Quote: Wizard

Quote: DJTeddyBear

However sometimes it can be done. For example if part one is a regular bingo, and part two is a postage stamp, either winning combination can occur first and on rare occasions it can occur at the same time, and even on the same board. link to original post

A single bingo and postage stamp do not completely overlap each other, thus I would expect the player to get paid for both.

I still don't understand what is wrong with my initial answer that Don challenged.
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The original post said this:

"how can we calculate the RTP/probability of getting one of the 4-corners in 30 draws when getting the coverall in 26 draws effectively "blocks" some of the times the 4-corner pays out."

I read this as asking for the probability of getting 4 corners in 30 or fewer without simultaneously getting the coverall in 26 or fewer.

I don't know if this question will throw a monkey wrench into the discussion or not, but...

Are these questions for a live bingo game, or for Class-II slots that are bingo based?

On a side note, do bingo based slots pick a specified number of 'balls' and then look at the results to then map it to the slot result? Or merely keeps picking (to a predetermined maximum), stopping at the first winning combination? I suspect the former.

The only reason I bring this up at all, is because it's kind of impossible to have a coverall in X numbers, without also having the 4 corners at that time - and probably gets the corners a lot sooner.

Of course, getting a coverall in 26 numbers is rare.

And if we're talking about a live bingo game, more likely than not, a different player (or board) is going to get the 4 corners (or other preliminary pattern) and not the player/board that gets the coverall.

Quote: DJTeddyBear

On a side note, do bingo based slots pick a specified number of 'balls' and then look at the results to then map it to the slot result? Or merely keeps picking (to a predetermined maximum), stopping at the first winning combination? I suspect the former.
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I have seen both.

It will be at least a few days until I can visit a Class II Gaming Facility to reconfirm.

There are LOTS of bizarre ways to map bingo results to a slot machine prize. Those help screens are absurdly interesting, especially the ones that are 20 pages of numerical tables mapping ball products to prize awards.