The Odds on Bingo

Today, having plenty of better things to do, I decided to amuse myself by calculating the odds involved in the game of bingo. Specifically, I am considering the standard US version of the game, in which the numbers run from 1 to 75, the cards use a 5 x 5 grid with the central cell marked as “free,” and the winner is the first player to complete a horizontal, vertical or diagonal row of five marked cells.

What, I wondered, are the chances of two or more players getting bingo at the same time and having to share the prize?  Well, in order to calculate this I first had to find the chances of a single player getting bingo as each number is called. This can happen after as few as 4, or as many as 71 numbers have been called, although these are both very remote possibilities: only about 1 chance in 300,000, and 1 chance in 700,000 respectively. Ninety-nine percent of the time, the bingo will come somewhere between calls 14 and 62, with the most likely time being on the 43rd call (about a 1-in-25 chance).

Armed with these figures I was able to calculate the chances of sharing a prize. This of course depends on the number of players in the game, but not as much as you would expect. (With more players, there is a better chance that one of them will go bingo fairly early on in the game, at a point when the chances of any other individual player winning are still quite small.)

No. of players

Chance of 1 winner

Chance of 2 winners

Chance of >2 winners

1

100.00%

0.00%

0.00%

2

97.17%

2.83%

0.00%

3

95.96%

3.95%

0.09%

4

95.21%

4.62%

0.17%

5

94.67%

5.10%

0.23%

6

94.24%

5.47%

0.29%

7

93.89%

5.77%

0.34%

8

93.60%

6.02%

0.38%

9

93.34%

6.24%

0.42%

10

93.11%

6.43%

0.46%

15

92.24%

7.16%

0.61%

20

91.62%

7.66%

0.72%

30

90.73%

8.38%

0.90%

40

90.07%

8.89%

1.04%

50

89.55%

9.29%

1.15%

100

87.83%

10.59%

1.58%

150

86.73%

11.39%

1.88%

200

85.90%

11.98%

2.12%

This table shows that even with up to 40 players, there’s less than a 1-in-10 chance that the prize will have to be shared (and an even smaller small chance that it will have to be split among 3 or more players). Bingo is obviously well suited to being played by groups of different sizes.

Another attraction of bingo is that most non-winners feel they came very close to winning. In order to calculate the figures above, I looked at all possible configurations of cell markings on a card (well, my computer did the actual looking, as there are a total of 2^24, or 16,777,216 possibilities). Of these, there were 10,624,010 configurations that a non-winning player might hold (those that did not include a row of five marked cells).  And 8,950,584 (over 84%) of these non-winning combinations were “pregnant,” that is, they had one or more rows with exactly four marked cells. As a result, most players in a game will be only one number short of having a winning card, before one lucky player calls “bingo”. This not only increases the excitement of the game, it also encourages repeat play.  No wonder bingo is so popular!

How fair is the World Cup?

With England through to FIFA World Cup semi-finals, I am wondering: how fair is the knockout system that determines the first, second and third-place winners?

To simplify things, let us assume that the teams can be ranked in order of their relative strengths, and that each team plays consistently to its ranking, so that a higher-ranked team will always beat a lower-ranked one. (Yes, I know this is a gross oversimplification.) Let us also assume that the seeding system used by the organizers is fair, so that the eight seeded teams (which include the host nation) are the top ranking eight teams (another big assumption).

These 8 seeded teams were then assigned randomly to 8 groups and based on our assumptions, each seeded team could be expected to win its group. It would then meet a different group’s (unseeded) second-place team in the round of 16, which it could also be expected to beat. So, based on our assumptions, the quarter-finalists should be the top-ranking eight teams.

In the knockout competition, the winning finalist is awarded first place, the losing finalist second place, and the losing semifinalists compete in a playoff for third place. So using this system, the #1 ranked entrant will always get first place.  But, based on the random way the teams have been bracketed together, what are the chances of a team winning second or third place?

Well, I did the math, and here are the chances for each team’s possible outcomes:

Team rank 1st place 2nd place 3rd place Playoff loser Eliminated in quarter-finals
#1 1 0 0 0 0
#2 0 4/7 2/7 0 1/7
#3 0 2/7 3/7 0 2/7
#4 0 4/35 8/35 8/35 3/7
#5 0 1/35 2/35 12/35 4/7
#6 0 0 0 2/7 5/7
#7 0 0 0 1/7 6/7
#8 0 0 0 0 1

As you can see, the knockout system is only accurate for determining the first-place winner and there is a very good chance that the second and third places won’t go to the “right” teams.  There’s even a chance that the second-place winner could be the #5 ranked team!

In fact, there’s only an 8-in-21 chance that the #2 and #3 teams will both be placed correctly, a 4-in-21 chance that their places will be reversed, and a 3-in-7 chance that one or other of them won’t get placed at all.  The only certainty is that at least one of these two teams will get second or third place.

Note that this analysis doesn’t take account of the random factors that affect every team’s match-day performance: it only reflects the advantages and disadvantages that can arise from the “luck of the draw.”

Conclusion: the World Cup’s knockout system is not very fair at all when it comes to allocating second and third places.

DISCLAIMER: I’m posting this before England’s semi-final game is played, so don’t consider this to be sour grapes!