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Olympic game theory

Today, London 2012 organisers will start taking your cash for tickets to next year's events. Tony Crilly calculates how likely you are to have spent a fortune

As of this morning, the organisers of next year's London Olympics will begin to extract money from the bank accounts of those who applied for tickets.

The only problem is that for many who have gambled in over-applying for tickets for sought-after events such as cycling and athletics, sums of thousands of pounds could be taken from their debit or credit cards. Or they could have no money taken out.

So how likely are you to have got the tickets that you want for certain events?

Can mathematics help you with this uncertainty? Surely mathematical uncertainty is an oxymoron. After all, mathematics is billed as the subject of the utmost certainty – what could be more definite than 2 + 2 = 4 and "evermore will be so"? Traditionally, maths is the subject of the black and white. There is "Yes" and there is "No" but not "Maybe".

This view would be to ignore the wonders of the mathematics of probability.

The distribution of Olympics tickets works this way: there are a certain number of tickets, N = 10,000 (say), available for distribution. You then apply for a ticket. If the session is undersubscribed, where fewer than 10,000 people apply, you will get a ticket. If more than 10,000 people apply, your name goes into a hat and names are drawn out at random. So if 100,000 people apply, your chance of getting a ticket is one chance in 10, or using percentages, 10 per cent.

This way of distributing tickets is the "level-playing-field model", and you can't improve your chances, because only one application is allowed. Everyone has the same chance or probability of getting a ticket.

With a single ticket, the chance of winning the top prize in the UK Lotto is one in 14 million, which is a minuscule 0.000007 per cent. In the case of lotteries, you can improve your chances by buying more tickets. If you buy "x" tickets you multiply the probability by "x", but you would have to buy about 1.4 million tickets to give yourself a 10 per cent chance of winning the top prize. So there's a fair chance of getting an Olympic ticket, and a slim chance of winning the lottery.

At least the way of distributing tickets for the Olympics is not a "first come, first served" model along the lines that airlines use to sell their seats. Here the airline plays a mathematical game with travellers. The airline sets a price but as the aircraft fills and their costs are covered, the airline can afford to raise the price. With a few days to go it can then lower the price, recognising that an empty seat means zero revenue. On the other side of the game, the passenger wants to travel but does not want to pay a high price – so it's a measure of judgement when to purchase the ticket. The organisers of the Olympics aim to fill all the seats but will not vary the price of a ticket once it is announced.

The advice given to Olympics customers to spread their risk is sound. But if you applied for a number of tickets in the hope of getting a few, you might be on the end of a hefty bill. You might expose yourself to £2,000 of risk (say). You would have little to worry about if you applied for events for which there was a high demand. If you reckoned the chance of gaining a ticket to be 5 per cent, your expected outlay would be 5 per cent of £2,000, which is £100. This is a gauge figure which enables decisions to be made. On the other hand, it would be dangerous to allocate £2,000 to only events of low demand. In that case, the probability of getting a ticket would be high and the expected outlay much higher.

In the case of the Olympics you have to surmise the probabilities but it is so much better than a blind hunch.

Unlike guessing who's going to be next year's 100m champion, buying Olympics tickets, to some extent, puts us into the realm of the unknown.