Wizard of odds bingo
(Math notation is generally the same as that used in Microsoft’s Excel. The MathNotation link will also give examples of the notnotation as used here.)
For 90 number Bingo ( 3 rows, 9 columns) please see Bingo 90.
The Bingo Statistics link gives tables and graphs showing the probability for getting “Bingo” after the announcer has called “N” numbers. Tables and graphs cover both a single board and a 50 board game.
The Bingo 4 Corners and Letter “X” (both diagonals) link shows the single board probability of getting these patterns after “N” numbers have been called.
The Bingo Picture Frame (all 4 edges) and Letter “Y” link shows the single board probability of getting these patterns after “N” numbers have been called.
The Bingo Probabilties for a Complete Cover link shows the probabilities for covering all squares on a Bingo board.
The “How to calculate” link shows how to calculate these numbers including how to calculate the probabilities when any arbitrary number of boards are being played in a game.
Probabilities for Swedish Bingo. A Swedish Bingo card has the familiar 5 rows and 5 columns, but the middle cell is not free. It has to be filled by having its number called.
Rules of the game: A typical Bingo card has 24 semi-random numbers and a central star arranged in a square of 5 rows and 5 columns. A Bingo card might look like:
We used the phrase “semi-random” to describe the numbers because the numbers in each column are confined within ranges. Column 1 will contain 5 random numbers in random order, but they are within a range of 1-15. Similar ranges exist for the other 4 columns (16-30, 31-45, 46-60, and 61-75). The central location is a “Free” spot. There are (15!/10!)^4 * (15!/11!) = 5.52+ E26 (more than 552 million billion billion) possible combinations that could exist – any one of which would be a legal Bingo card. (The “!” symbol is the mathematical notation for Factorial. e.g. Factorial(5) = 5 * 4 * 3 * 2 * 1 = 120.)
Note: The above number of combinations assumes the numbers in any column can be in random order. If the numbers in any column are always in sorted order with the lowest number on row 1 and the highest number on row 5, then the number of combinations in each of 4 columns is reduced by 5! = 120 and the number of combinations in the center column is reduced by 4! = 24.
Initially, the central “*” is counted as a “free” or “called” cell. Then, an announcer will call out numbers selected randomly within the total 1-75 range. (Usually this is done by randomly removing numbered balls from a revolving drum.) Whenever one of these called numbers matches a number on a player’s Bingo card, the player marks that number as “called”. Eventually, there will be a straight line of 5 called numbers that fill a row, fill a column, or form a corner-to-corner diagonal line. (Note: the “Free” center space can be part of the straight line). At this point the player yells “Bingo” and the game is over.
Probabilities: Of interest, in a single board game – What is the probability the player will have a “Bingo” after the announcer has called “N” numbers? Also, in a multiboard game, what is the probability that the first “Bingo” will show up after “N” numbers have been called? (Check the Bingo Statistics link.)
Variations on Bingo: Other patterns can be used for the game of Bingo. For example, a winning Bingo could be defined as filling a 2×2 block anywhere on a Bingo card. There are 16 possible locations where a 2×2 block could be located. Other Bingo variations could include filling any of the 9 possible 3×3 blocks, or filling a 2×3 block. A 2×3 block could also be rotated for 24 possible winning “Bingos”.
Other Bingo websites: The “Wizard of Odds” also has bingo statistics information – especially the gambling aspects of Bingo as well as a lot of good stuff on gambling in general. The probabilities given here match those in the “Wizard’s” tables. (It’s reassuring to have two independent calculations come up with the same results.)
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Probability Analysis that you will have a "Bingo" after "N" numbers have been called.
Recommended online casinos
On March 16, 2020, Las Vegas and the entire country was getting ready to hunker down. Many already had hunkered, some had little to no fear, but most people seemed to be somewhere in the middle. The Wynn and all MGM properties were scheduled the shut down the next day. In the news that day, the stock market set a new record for the largest point drop (DOW dropping almost 2,997 points or 12.93%) and the White House discouraged gatherings of more than ten people.
Perhaps it wasn’t the smartest decision I ever made, but on that day I decided to see what a Las Vegas casino looked like, in particular the Suncoast, a mainly locals casino near where I live. In particular, I thought there would be little competition in the bingo room. In Las Vegas, the prize pool in bingo is fixed. Thus, the fewer cards you’re competing against, the better your odds. In bingo, the value is proportional to the ratio of the prize pool to the number of cards in play.
When I arrived at about 2:30 PM, the parking lot was about half as full as it normally would be on a Monday afternoon. Once inside, I noticed every other slot machine was turned off. The table games had strictly three chairs per table. In a casino with about 50 table games, only about 10 were open, with about half standing dead. The 64-lane bowling alley had only four games going. I was hoping this pattern would continue to the bingo room, but, alas, it seemed to have about 75% of its usual number of players for the 3:00 session. This is a rough estimate, as I hadn’t played bingo in several years and even then I preferred the 9:00 PM session.
I came with plenty of money, hoping to find a nearly deserted room in which case I would be a card hog and monopolize the game. However, based on about 60 players in the room, I figured wasn’t going to be an outstanding value. Perhaps not even any value at all. It would depend on how many cards my competing players were buying. My view was to spend a little money for research purposes to estimate the number of players and maybe play more sessions if I detected a good value.
I thought my initial selection was cards would come to $54, enough to get a free Large Rainbow pack. However, there was some kind of other promotion running and my total came to $44 only. So I purchased three $2 Double Daub cards, a special progressive game for game 8, that I previously considered a bad value. My other $44 purchased 120 regular cards of varying colors. Let me add that, no, I did not validate. I never do. If the Cash Ball jackpot were high enough for validating to be a good deal, I wouldn’t play at all, as it would induce too much competition.
In games 1 to 7 I was “cased” many times, meaning one number away from a winning. However, alas, the numbers I needed were never called. When game 8 came around I took a break from taking notes and drew out a map of the 50 states in my notebook (something I have a good skill for). Half way through, I was surprised when my device to play electronically beeped, which means I was one number away from winning. One or two more balls were called and I nobody had won yet. Then whatever number I needed, I think something in the 20’s, was called! I shouted out “bingo!” enthusiastically. However, the caller didn’t seem to acknowledge me. Another player, an elderly woman, said in a somewhat annoyed voice, “You have to raise your hand.” Boy, was I rusty at bingo. So I waved both hands frantically. If another number had been called, it would have voided my win. An attendant did get to me, went through the long verification process, including calling out every number that contributed to the win. It verified, of course, and I won a progressive jackpot of $1,135! This was truly dumb luck. It would be like me making a side bet in a table game (something I highly advise against doing), once, and hitting a 200 to 1 shot. The rest of the game I continued to be cased often but never won.
When I got home I used my bingo calculator to estimate the number of cards in play. I won’t bore you with all the details, except to say the bottom line was there were about 5,000 regular cards in play. This would suggest the average player had about 85. Assuming the player purchased unvalidated B Electronic Specials, the best value, he would have had about a 93% expected return, or a 7% house edge. Please take this estimate with a grain of salt, as that 5,000 figure is just an estimate. I would say the margin of error is about 1,500 cards. One session is also a bad estimator of bingo value in general.
After leaving bingo victorious, I went to the sports book to see if there was anything to bet on. The room was empty except for one bored writer sitting at an empty window. The entire board of bets was filled with futures. No specific games to bet on, except one. It was a bet on a Turkish league soccer game. In a very un-Wizard like move, I threw $20 down on the underdog, Caykur Rizespor to win at +330 odds, to be played on Saint Patrick’s Day. The luck* of the Irish was not with me during the game, as my team lost 0-2.
Suffice it to say I probably won’t be back for bingo, at least for Coronavirus reasons. For one reason, there are still too many elderly people risking death to play. Remember, in bingo the enemy isn’t the casino, it’s the other players. For another, I don’t want to die either.
* Let the record show, I was just joking. I don’t believe in luck, as defined as a force, other than in a pure mathematical sense, that causes good or bad things to happen.
The Wizard investigates playing bingo during a near