Bang or No Bang Game 3: Who Wants to be a Millionaire?

On ITV's "Who Wants to be a Millionaire", contestants are asked questions with four possible anaswers. The more questions you answer correctly the more money you win. If you are able to answer fifteen questions correctly in a row, you will win one million pounds! To help you in answering the questions you are given three life-lines which you can use at any point:

• 50:50 - the computer will remove two of the wrong answers, leaving the right answer and one remaining wrong answer
• Ask the audience - the entire studio audience is asked to vote for what they think is the right answer
• Phone a friend - one person from a pre-selected set of friends is telephoned and allowed 30 seconds to answer the question

If you are certain of the answer then probability doesn't enter in to it. But what if you have have no idea what the answer is? Should you just guess?

There are times when it is fine to guess, as you won't risk losing any money (for example when you reach £1000 and £32000, these are "safe" question. Once reached, you are guaranteed to win at least that amount).

But what if you get all the way to the fnal question, and have no idea what the right answer is? Should you guess? In this instance you will have answered 14 questions correctly and you will be sitting on £500 000. If you guess the next question and get it wrong, you will lose £468 000, dropping back to the last "safe" amount, or £32 000. If you get it right you will win £500 000, taking the £1 million jackpot.

The probability of you guessing the right answer, assuming it's a random guess, is one-in-four, that is

P = 1/4 = 0.25

You can increase the probability of being right if you decrease the number of ways you can be wrong. This is where the 50:50 lifeline comes in. Use of this life-line will increase the probability of you guessing correctly to

P = 1/2 = 0.5

Even still, you are still just as likely to be wrong as right. You can increase your chance of being right by asking the audience or by phoning a friend, but they might not know the answer either.

You can use mathematics to work out whether you should guess!

Given that you have used 50:50 and have an equal number of ways of being right (1) as being wrong (1), guessing would be just like tossing a coin. But what is at risk in this high-stakes game of heads or tails?

Let's say that if you throw heads you win, tails you lose. If you throw heads you will win £500 000. If you throw tails you will lose £468 000 (This is the £500 000 that you currently have minus the £32 000 that you are guaranteed to win whether you get the question wrong or not).

This means that the odds are "in your favour" to make a random guess. That is you stand to win more than you stand to lose. But would you? Most people wouldn't.

If you were to play this game a hundred times then you'd be right to guess, as on average you'd win more than you lose. But you only get one chance in this game, and the utility of the money at stake means that you should just take the £500 000 and run.

Utilty is the usefulness of something. The usefulness to most people of £500 000 is not that much different from the usefulness of £1 million. That is, both amounts would change your life.

The utility of £32 000 is high but far lower than the utility of £500 000. You could do a lot with £32 000 but not nearly so much as you could do with £500 000.

This introduces the concept of risk. Risk doesn't just have to do with the probability of an event happening, you also have to consider what you might lose if it doesn't happen.