Bingo - "x Marks" expected calls

Hi Mike

I have enjoyed your site for many years (decades now).


On your Bingo Probabilities 2 pages, you say (if I understand correctly) that (one-way only) patterns containing the same number of marks have the same expected calls.

But, don't you have to take into account the fact that some "x Mark patterns" omit some of the columns - and, therefore, reduce the number of balls eligible to be called.

So, for example, on your Bingo Pattern Probabilities pdf, you show two 4-mark patterns (Inside Corners and Large Diamond Corners) having different expected calls. I'm pretty sure the different expected calls calculation is due to different numbers of columns being used for calling.

Or, have I just misunderstood your statement on Bingo Probabilities 2?

Can you quote some specific numbers, as I'm not sure I see you example. For the benefit of others, here are some pertinent links.

Bingo probabilities 1
Bingo probabilities 2

About half-way down on Bingo Prob 2 is a table titled "Expected Calls to Cover Pattern of x Marks".
Using that table we see that, for any one-way pattern of 4 marks, the expected calls is 60.8.
Yes, that is for only one card in play. But the implication is that for any n cards the expected calls should be equal.

Then in your pdf at /pdf/bingo-probabilities.pdf,
you show 4 CORNERS averaging 8.43 calls (for 300 cards), but the LARGE DIAMOND CORNERS pattern average is 11.86.

Both of those patterns are 4-mark patterns - so, if I read Bingo Prob 2's implication correctly, both expectations should be equal ... but, they aren't.

Further, I'm speculating that the unequal expectations (in the pdf) are due to the fact that the 4 CORNERS pattern omits 3 columns, leaving only 30 eligible balls to be called ... and the LARGE DIAMOND CORNERS pattern omits 2 columns, leaving 45 eligible balls to be called.


Thank you very much.

Okay, now I understand the question. Let me try to rephrase it for the benefit of other readers.

We only need to concern ourselves with this document: Bingo Pattern Probabilities.

Let's just look at the case of 300 cards.

For the four corners pattern, it shows an average number of calls for somebody to win of 8.43 and an average number of winners of 1.39.

For the Large Diamond Corners pattern (if we think of the card like a 5x5 Battleship board, it would require marks on A3, C1, C5, and E3), it shows an average number of calls for somebody to win of 11.86 and an average number of winners of 1.29.

Both patterns have just one way to win and four specific positions on the card to cover. Why are the average number of calls and winners not the same?

The answer is a bit hard to explain, but those who have played a lot of bingo will probably get it.

The distribution of balls called by letter is not necessarily uniform. There will usually be a surplus of some letter and a shortage of others. The Four Corners pattern requires numbers in two columns only, and the Large Diamond Corners in three. The difference is the N column. If there is a shortage of N numbers called, then it's going to take the Large Diamond Corners to hit.

In general, patterns that require fewer letters to be covered tend to hit sooner and ones that require more columns tend to hit later.

I know this isn't my best explanation ever. Let me ruminate on it for a possibly better explanation. Or anyone else may chime in.

If you are willing to say there is a difference in the expected calls for 300 cards for these two 4-mark patterns, shouldn't there also be a difference in the expected calls for 1 card? But, your "Expected Calls to Cover Pattern of x Marks" says that ALL 4-mark patterns have the same expected calls.
That's the (apparent) contradiction I was trying to point out.

In my view, it's easier to see the reason for the difference if you recall how Bingo is called.
For patterns that omit a column, balls for that column do get DRAWN ... they just aren't CALLED. (Ineligible drawn balls are silently recognized by the caller and simply put aside and the caller moves ahead. And, yes, sometimes this makes for an unpleasant delay in the "action".)

Functionally, this is the same as if they loaded the hopper with only the 30 eligible balls for the 4 CORNERS, and the 45 eligible balls for the LARGE DIAMOND CORNERS. And so, with less eligible balls to draw from, you naturally get a smaller number of expected calls for 4 CORNERS (8.43) than for LARGE DIAMOND CORNERS (11.86).

If all that's true, then the table "Expected Calls to Cover Pattern of x Marks" is not correct in its claim that all x-mark (one-way) patterns have the same number of expected calls.
(It would be correct if you change it to "Expected DRAWN Balls to Cover Pattern of x Marks" ... but that wouldn't be of much interest, I think.)

Quote: BobThePonderer

If you are willing to say there is a difference in the expected calls for 300 cards for these two 4-mark patterns, shouldn't there also be a difference in the expected calls for 1 card?

No. For one card the probabilities are the same.

If The Wizard will indulge me just one more time ... a thought experiment, please.

Let's say the winning Bingo Pattern is the N column. Since the Free Space is irrelevant, that would be a 4-mark pattern.
Let's also say there is only 1 card in play.

Your table says the expected number of CALLS would be 60.8.

But, 60.8 is greater than the total eligible balls. The caller will only CALL N31 thru N45 ... a total of 15 balls. Non-N balls DRAWN will not be called ... they will be (silently) put aside.

So, as I tried to point out earlier, I think your table holds true for EXPECTED BALLS DRAWN, but not for EXPECTED BALLS CALLED.


If you still disagree, I will humbly admit to defeat and promise to not try to make my point again.


All the best,
Bob

Quote: BobThePonderer

If The Wizard will indulge me just one more time ... a thought experiment, please.

Let's say the winning Bingo Pattern is the N column. Since the Free Space is irrelevant, that would be a 4-mark pattern.
Let's also say there is only 1 card in play.

Your table says the expected number of CALLS would be 60.8.

But, 60.8 is greater than the total eligible balls. The caller will only CALL N31 thru N45 ... a total of 15 balls. Non-N balls DRAWN will not be called ... they will be (silently) put aside.

So, as I tried to point out earlier, I think your table holds true for EXPECTED BALLS DRAWN, but not for EXPECTED BALLS CALLED.

If you still disagree, I will humbly admit to defeat and promise to not try to make my point again.

All the best,
Bob

That 60.8 counts B, I, G, and O balls. They will still be called. A ball is never silently put aside.

Thank you. We can finally see where we differ.

In the casinos and the bingo halls around here (Louisiana and Arkansas), any ball drawn that is not a possible hit is indeed put aside (as I described previously) and is not called..
This is done (I believe) to make the game go faster, since an experienced caller can recognize these non-playable balls and quickly move on.

In fact, if the caller mistakenly does call a non-playable ball, the crowd will let him or her know right away.


I appreciate your time with this, Mike.

Quote: Wizard

Let's just look at the case of 300 cards.

For the four corners pattern, it shows an average number of calls for somebody to win of 8.43 and an average number of winners of 1.39.

For the Large Diamond Corners pattern (if we think of the card like a 5x5 Battleship board, it would require marks on A3, C1, C5, and E3), it shows an average number of calls for somebody to win of 11.86 and an average number of winners of 1.29.

Both patterns have just one way to win and four specific positions on the card to cover. Why are the average number of calls and winners not the same?

The answer is a bit hard to explain, but those who have played a lot of bingo will probably get it.

The distribution of balls called by letter is not necessarily uniform. There will usually be a surplus of some letter and a shortage of others. The Four Corners pattern requires numbers in two columns only, and the Large Diamond Corners in three. The difference is the N column. If there is a shortage of N numbers called, then it's going to take the Large Diamond Corners to hit.

In general, patterns that require fewer letters to be covered tend to hit sooner and ones that require more columns tend to hit later.

I know this isn't my best explanation ever. Let me ruminate on it for a possibly better explanation. Or anyone else may chime in.

I found this to be fascinating. I had to think about this for a while. Here's a way to illustrate the point you are explaining:

Consider a game with an infinite number of cards in play (or in which every possible bingo card is in play.). The Four Corners game will be over the moment they draw 2 numbers from B(1-15) and 2 numbers from O(61-75). However, the Diamond Corners game will be over as soon as they draw 1 number from B(1-15) and 2 numbers from N(31-45) and 1 number from O(61-75). Combination math can now quickly explain the difference in the average number of cards drawn with these patterns (for the case of infinite bingo cards). For non-math people, you can explain that the two patterns have different numbers of the 75 possible balls that cannot possibly contribute to someone winning (which is pretty much what you said.)

What's also evident is that a row pattern (say top row) and a column pattern, (say B column) both require 5 numbers but will have different average game lengths (as measured in numbers drawn) whenever two or more cards are in play - and the average game length will become progressively more different for the row and column patterns as the number of cards in play is increased.

Quote: gordonm888

Combination math can now quickly explain the difference in the average number of cards drawn with these patterns (for the case of infinite bingo cards).

Thank you.
Your idea to use an infinite population of cards as an explanatory device is very interesting.

Would you please take it a bit further and show the combinatorial math that quickly explains "the difference in the average number of cards drawn with these patterns"?

Yes, I like gordon's infinite card explanation too.

I've been thinking about this more. Suppose the game is drawn WITH replacement.

What would be the expected number of balls drawn to get both 4 corners and the Diamond pattern (two separate questions).

Here are the results of a simulation WITHOUT replacement of the average number of balls needed to call, assuming infinite cards.

Four corners: 12.8289
Diamond: 11.3645

An idea:

Following up on Gordon's -

Instead of infinite cards, how about all possible 3 x 3 cards, giving 8! = 40,320 possible cards. Would that help make the combinatorics tractable?

Or, I'm thinking maybe even a brute force spreadsheet listing could show how more columns affect the expected calls. (Should allow exact answers, yes?)

For example:
Let's say for the 1st case (Pattern 1), our 2-mark (single column) Bingo pattern is (using Mike's battleship notation)
A1
A2

Then, for the 2nd case (Pattern 2), also a 2-mark (but, needing two columns)
A1 C1

So, now would it be easy to show that Pattern 1 tends to hit sooner than Pattern 2?

Quote: BobThePonderer

Instead of infinite cards, how about all possible 3 x 3 cards, giving 8! = 40,320 possible cards. Would that help make the combinatorics tractable?

In a 3 x 3 card, assuming a free space in the center, how are you getting 8! different cards?
Are you saying that each of the numbers 1-8 appear on every card?
In that case, I don't see how A1 A2 is more likely than A1 C1.

Or by "3x3" card, are you limiting it to 1, 2, 3 in column 1, 4 and 5 (and the free space) in column 2, and 6, 7, 8 in column 3?

Yes, thank you, Don, for catching my error.

My idea (an extension of Gordon's idea of infinite cards -or- all-possible cards, I think) of a smaller 3x3 analogue of the regular 5x5 bingo card, numbered as you say, would have only 6 x 2 x 6 = 72 possible cards in the complete set.

So, now that that is corrected, shouldn't having such a small set of possible cards make it easier to show (using combinatorics or brute-force listing) how the two patterns have different expected calls - due to Pattern 1 being confined to one column and Pattern 2 needing two columns?


Thanks
Bob

Quote: BobThePonderer

Yes, thank you, Don, for catching my error.

My idea (an extension of Gordon's idea of infinite cards -or- all-possible cards, I think) of a smaller 3x3 analogue of the regular 5x5 bingo card, numbered as you say, would have only 6 x 2 x 6 = 72 possible cards in the complete set.

So, now that that is corrected, shouldn't having such a small set of possible cards make it easier to show (using combinatorics or brute-force listing) how the two patterns have different expected calls - due to Pattern 1 being confined to one column and Pattern 2 needing two columns?

I did a brute force check on all 40,320 permutaions of balls numbered 1-8, assuming all 72 possible cards were in play for each of the games.
Assuming the object of the game was to get either the top left and bottom left corners, or the top left and top right corners, the results were:
15,293 games had the left column corners win
12,187 games had the top row corners win
12,840 games had a left column corners win and a top row corners win at the same time

Quote: ThatDonGuy

I did a brute force check on all 40,320 permutaions of balls numbered 1-8, assuming all 72 possible cards were in play for each of the games.
Assuming the object of the game was to get either the top left and bottom left corners, or the top left and top right corners, the results were:
15,293 games had the left column corners win
12,187 games had the top row corners win
12,840 games had a left column corners win and a top row corners win at the same time

Thanks, Don.

But, I believe your result is for a bingo with EITHER pattern being a winner ...

The original question was concerned about the difference in expected calls for a Pattern 1 game versus the expected calls for a Pattern 2 game.

Can your brute-force attack be applied to each of your patterns separately ... and yield the expected calls for each one?

Given all 18 different cards playing simultaneously over all 40,320 possible permutations of the 8 numbers, I get:

An average of 5.82386 draws to get the two corners in the left column

An average of 5.84846 draws to get the two corners in the top row