Bingo Odds

"How big are my odds to win a Bingo?", is a question that every player asks himself before buying a ticket or before deciding how many tickets is needed to improve odds.

To answer this question we have to explain the basic rules of the game and see how they influence the odds.

In Bingo, participants are equipped with one or more cards, each intricately divided into squares. These cards are structured with five rows and five columns.

The columns are sequentially labeled as "B", "I", "N", "G", and "O", moving from left to right. The central square holds the special designation of being the "free" space. Within column B, the five squares feature distinct numbers chosen from the range of 1 to 15. Moving to column I, there are five numbers within the range of 16 to 30. Column N accommodates four numbers from 31 to 45. The subsequent column, G, incorporates five numbers spanning the range of 46 to 60. Lastly, column O is composed of five numbers ranging from 61 to 75.

This structured layout sets the stage for the engaging game of Bingo, where players must mark off numbers on their cards as they are called out, aiming to achieve specific patterns for a winning outcome.

There are a few ways of winning a bingo, from covering just one line, two lines or making simple patterns like diagonal to covering the full card which is called blackout or full house.

The game is played until someone covers the whole card, so obviously the number of participants will tell you how big your chances are to win the game in the end. But beware that the participants can but multiple tickets, which means that the total number of tickets sold will show you the chances of winning a bingo:

Probability of winning a game = Number of tickets bought by player / Total number of tickets sold

This means that if we have 10 players and each buys one ticket, everyone has 1/10 chances, which is exactly 10%, of winning a game. But if one player decides to buy 11 tickets and others buy one ticket each, total number of tickets is 20, and the chances for the person with 11 tickets are 11/20, which is 55%, while the other players have chances of 1/20 or 5% of winning a game.

Of course the investment of the player with more tickets is 11 times higher than the investment from other players.

The jackpot will be different for these two situations since double the amount of tickets in the second game will mean that the jackpot will also double its size. Here we come to the interesting situation in which the player with just one ticket in the first game will win the jackpot 10 times bigger than the investment. On the other hand in the second game the same player had improved their chances, but also will also win just double the amount invested.

Number of Cards

As we saw previously, the amount of bingo cards bought will improve our chances, but how much the chances will be improved depends on the total amount of cards in play. If 1000 cards are bought by players, it is not much of a difference if we buy one, two or three cards, since the chances will be 0.1%, 0.2% and 0.3% for those situations.

But let’s see in some more detail the dependence between the number of cards and number of calls needed to win a bingo.

An announcer randomly selects numbers ranging from 1 to 75, announcing each one as players mark the corresponding grid square on their cards. In standard Bingo, the objective is for a player to be the first to mark an entire row, column, or diagonal, with a total of 12 winning configurations. Eight of these configurations exclude the center "free" space, while four involve it (column N, row 3, and the diagonals).

Single Line Pattern

This pattern necessitates players to daub five adjacent numbers on their bingo cards, forming a line vertically, horizontally, or diagonally. The single-line pattern is commonly the easiest to achieve in bingo, boasting a probability of someone winning in the first 41.37 calls.

Double Line Pattern

In this pattern, players aim to daub two complete lines of five numbers, whether diagonally, vertically, or horizontally. On average, a player is expected to win a double-line pattern game in 50 numbers or calls.

Four Corners

For this pattern, players must daub all four numbers situated in each corner of their bingo cards. With only four possible numbers to mark off, the odds are more extensive, and typically, a player secures this pattern on the 74th or 75th ball.

Blackout / Full House

A blackout entails a player crossing off all 24 numbers on their bingo card. However, winning a blackout before the first 15 calls is deemed impossible. In a game of 75-ball bingo, it usually takes approximately 72.96 calls for a player to achieve bingo on a single card.

Be aware that here we talk only about the case where you have one card bought, and with more cards, more calls will be needed. That we will see in more detail in the coming chapter.

Calls

Here we will see the probability of achieving any bingo relative to the number of calls executed, again for one card bought:

Let’s see what happens if we buy 10, 50 or 100 cards and how it improves our chances of winning:

Number of Players

It is hard to calculate the probabilities only by the amount of the players engaging in the game since we don’t know how many cards each of them will buy. That is why we will count each card sold as if they were separate players. The number of the players is anyway irrelevant since the payout will be paid in equal amounts to every winning card.

Here are the anticipated numbers of calls required for someone to achieve any bingo, based on the number of players:

For the blackout bingo, things are slightly different. Since the amount of the combinations for calling is incredibly high and measured in trillions, there is very little difference when we increase the amount of players in the room. Of course with an extremely low number of players, full house or blackout will happen somewhere around the 70th-75th call, but numbers stabilize around the 65th call when we have more than 20 players.

One of the interesting facts that people often overlook is that the chances for getting a blackout in the first 24 calls even by having all possible cards are 1/232. It is because of the fact that each card line has 5 or 4 numbers from one range of numbers, and the balls are drawn randomly. Therefore, often it can happen that more than 5 numbers from one line are drawn in the first 24 calls, which already destroys your chances of winning in the first 24 calls.

Number of Games Played

The number of bingo games played does not directly improve your chances of winning in subsequent games. Each bingo game is an independent event, and the outcome is determined by chance. The probability of winning a particular game remains the same regardless of how many games you have played before.

However, participating in more games may increase the overall likelihood of experiencing a win over an extended period. This is because each game provides an opportunity to win, and playing more games gives you more chances to be successful. It's important to note that bingo, like many other forms of gambling, is primarily a game of luck, and no strategy or prior outcomes can influence the results of future games.

In summary, while playing more bingo games may offer more opportunities to win, each game's outcome is independent of previous games, and there is no direct impact on your chances of winning in subsequent rounds.