# 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 Draw 2/1 7/3 9/4 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.

Send your answers to This email address is being protected from spambots. You need JavaScript enabled to view it.. Explain to us how you came up with the solution, as we may publish a selection of correct entries (if received by 1 November 2021).

• 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).