Probability puzzle: A certain bet
For fans of the England soccer team, this summer’s European Championship ended in heartbreak and disappointment. But there is always next year’s World Cup to look forward to – assuming England can qualify.
The team made a decent stab at qualifying when they played Hungary in September 2021 and won the match 4–0.
Ahead of that game, bookmakers were taking bets on the three possible outcomes: a win for Hungary, a draw, or a win for England. The odds offered by three different bookies are shown in Table 1.
Table 1: Odds being offered on the Hungary v. England World Cup qualifying match.
|Bookmaker A||Bookmaker B||Bookmaker C|
|Win for Hungary||13/7||9/5||17/10|
|Win for England||15/8||19/10||33/17|
If you were to bet on a win for Hungary with bookmaker C, the odds of 17/10 tell you that you would win £17 for every £10 bet you place should the team be successful. So, let’s say you bet £20 on Hungary to win and they do: the bookie would pay you £54 – that’s your £20 bet returned plus £34 profit.
Now, imagine it is August 2021 – a month before the Hungary v. England game, so you have no idea who is going to win. But you do have £99 to bet with. Based on the odds given in the table, what bets should you place to guarantee yourself a profit? Remember, you don’t know the outcome of the game, so this cannot inform your betting strategy.
- Puzzle set by Michael Fletcher and published in the October 2021 issue of Significance.
- Try your hand at Michael's previous puzzle, "Higher or lower?" - and see here for the solution (magazine subscription required).