### Win or Lose?

A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. After 2n plays he has won exactly n times. Has he more money than he started with?

### Fixing the Odds

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

### Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

# Why Do People Find Probability Unintuitive and Difficult?

##### Stage: 2, 3, 4 and 5

Published November 2010,December 2010,February 2011.

This is quite an easy question to answer: because it is unintuitive and difficult. I have been doing probability and statistics for decades and now spend a lot of time explaining ideas to everyone from school students to the government Chief Scientist, and I still find some concepts tricky. The only gut feeling I have about probability is not to trust my gut feelings, and so whenever someone throws a problem at me I need to excuse myself, sit quietly muttering to myself for a while, and finally return with what may or may not be the right answer.

Nevertheless in spite of (or perhaps because of) this difficulty, I believe it is vital that students, and not just maths students, have at least some idea of how probability works in the real world, even if they will never grasp the full ghastly apparatus of permutations and combinations. A few of this month's problems require mathematical manipulations (such as using probability tree diagrams in At Least One...), but the primary emphasis is on developing a feeling for probability - how it relates to our natural sense of uncertainty but also has many odd and unintuitive properties that play themselves out in the real world.

Problems such as In the Playground and The Car That Passes introduce the idea that uncertainty is something that can be discussed and analysed, and that our uncertainty can depend on what we believe and what we know. Probable Words demonstrates the rich texture of language that is used to deal with uncertainty, even before introducing probability as a mathematical language.

Other problems manage to contain extremely subtle ideas within the wrappings of a game. Sociable Cards is a fun trick to play with friends, while Same Number! can be neatly adapted to any size group. I give 20 wipe-clean boards and marker pens to students standing in a circle and ask them to write down their own unique number between 1 and 100. Time and time again duplicate numbers are chosen: even if they genuinely chose at random there would be an 87% chance at least two would choose the same number, and since people are useless at choosing random numbers the odds are even more in my favour.

Both these problems are essentially based on one of the most common probability 'tricks' - that if there are sufficient opportunities even an apparently rare event is likely to happen. A good example is when a family has a baby who shares a birthday with two previous siblings: this story crops up almost every year in the UK newspapers. Assuming all birthdates are equally likely (and classes are quick to point out that this may not be the case) then the probability is $\frac{1}{365}$ $\times$ $\frac{1}{365}$ = 1 in 133000, even though newspapers continue to put in an extra $\frac{1}{365}$ to make 1 in 48,000,000 (which would be the chances of three children being born on a pre-specified date, not just the date on which the first child happened to be born). Apparently each year around 160,000 children in the UK are born as the third in a family, and therefore we would expect this event to happen around once a year, which it duly does.

The Derren Brown Coin Flipping Scam is another fine example of a rare event (10 heads in a row) that will happen if you try enough times. But it also nicely shows the difficulty of interpreting a piece of apparently remarkable evidence (he is filmed flipping a head 10 times in a row) without knowing what you are not seeing (the whole day's filming it took to get this shot). Last One Standing can illustrate the routine occurrence of remarkable events very well: with a large enough audience I have had people flip a coin and get 10 heads in a row on their first try.

The other standard lack of intuition concerns whether or not past experience affects the likelihood of future events, as discussed in Do You Feel Lucky?  If a coin has come up heads many times in a row, it is one type of gambler's fallacy to believe that it is now more likely to come up tails, since tails is 'due'. Of course if one doubted it was a fair coin it would make more sense to bet on another head, which I sometimes show by using a two-headed coin. This is related to What Does Random Look Like?, which rests on our (wrong) intuition that randomness should be somehow regular and balanced, and so we think it unlikely that a coin will flip 4 heads or tails in a row. And yet with 20 flips there is a 77% chance of getting such a sequence, which generally enables a real random sequence to be distinguished from a fake.

But some problems remain difficult to explain. Take the advice given on some lottery websites that a mixture of odd and even numbers should be chosen because few winning tickets have all odd or all even numbers. Classes quickly identify this as ludicrous, given that any particular ticket has an equal chance of occurring, but it is not easy to clearly describe the faulty reasoning.

What is not in the problems this month? No complex permutations and combinations. No use of probability in everyday risky situations, since this will feature in future months. And no Monty Hall, as although this can satisfyingly baffle people, I have no idea what anyone ever learns from it, except that probability is baffling.

To be honest, I remain baffled by probability, but know that the only way through to some clarity is by keeping cool, drawing a probability tree diagram, and using some mathematics rather than relying on gut feelings.

David Spiegelhalter
Winton Professor of the Public Understanding of Risk
http://understandinguncertainty.org/