Hello maths fans! It’s been a very busy semester for your favourite geek sheep: sorting out activities for undergrads in Innovative Learning Week, lecturing Y1 undergrads in Proofs & Problem Solving, organising business/academic networking events, doing an art/science exhibition, and running an exhibition at the Edinburgh International Science Festival. Hopefully now that I have some time, you can look forward to blog posts about all of these things.
Today’s post is about our science festival fun. We (the School of Maths) teamed up with the School of Chemistry and went for a CSI-themed activity.The premise was that a priceless Egyptian vase had been stolen from the Museum and the visitors had to work out whodunnit. Using chemistry they had to analyse fingerprints and blood samples, and use UV and infrared data to identify substances found at the scene of the crime. After deciding on their prime suspect, they came over to the maths section, which was the courtroom. Here they had to weigh up the probabilities and statistics and then decide on whether their suspect was innocent or guilty.
Just as in real life, we didn’t reveal who actually did it, because we often don’t know for sure. And actually, we hoped that (despite all the evidence) the visitors would vote ‘Innocent’ because the evidence certainly didn’t prove anything beyond all reasonable doubt.
I’ve had my heart set on doing something like this for a while because I wanted to publicise the great work that our Forensic Statisticians (Colin Aitken and Amy Wilson) are doing right here in Edinburgh. They are analysing the occurence of drugs like cocaine on banknotes to help the police decide when someone is really a drug dealer. Apparently (and don’t quote me on this) most £20 notes (like, over 80%) have got traces of cocaine on them, so the police need help in deciding when the notes have been involved in drugs crime or when they have just accidently been placed next to the ‘dirty’ notes in a shop till.
Colin has also appeared as an expert witness in a few trials and has helped to write books to educate judges and lawyers about statistics. Like the general population, judges and lawyers often have a very bad intuition about probabilities. But unlike the general public, their decisions can really affect people’s lives. The classic example is the Sally Clark case. An expert witness for the prosecution claimed that there was a 1 in 73 million chance that two cot deaths could happen naturally in the same family, and therefore that Sally must have murdered her children. Not only was this statistic wildly wrong (the actual figure is about 1 in 100,000) but the conclusion of guilt is also wrong. Neither side took into account the probability of her innocence: despite the unlikelihood of double cot death, double murder is (statistically) even more unlikely. Such a mistake is called the Prosecutor’s Fallacy. In Sally’s case, it led to her spending 3 years in prison for a crime she never committed and then committing suicide a few years after she was freed.
So anyway, the idea behind our science festival exhibit was to show people how bad they were at judging probabilities and to introduce the idea of Bayesian Statistics (which is behind things like the Prosecutor’s Fallacy). Have a go at these questions and see if you can solve them! Answers will be provided in the next blog post.
1) One of the most famous examples of conditional probability is called the Monty Hall problem, or the Car-Goat problem. You are on a gameshow, trying to win a car. You definitely don’t want to win a goat. There are three doors, behind which the host of the show has hidden 2 goats and a car. You choose the door which you think conceals the car. The host then opens a different door to reveal a goat. Finally, you get to choose: should you stick with your original choice of door, or should you swap? Or does it make no difference?
2) On very similar lines is the following queston. I flip two coins and tell you at least one of them is a head. What’s the chance that the other one is also showing a head?
3) In a lottery there are a 10 numbers in a bag and you win the jackpot if you correctly predict which 4 numbers get pulled out. What are the chances of winning the jackpot? What are the chances of predicting 3 out of the 4 numbers?
4) If 100 people each flip a fair coin 5 times, how many of them will we expect to flip 5 heads?
5) On the wall there is a calendar for 2012. Visitors to the museum put their birthday on the chart as they come in. After how many visitors do we expect to see the first shared birthday? (I.e. two people with the same birthday.)
6) An eyewitness, Betty, says she saw a suspect leaving the scene of the crime, and that the suspect was wearing a hat. Betty is shortsighted and only correctly identifies hats 2 out of 3 times. That is, 1 time out of 3 she will think that someone is wearing a hat when they aren’t, and 1 time out of 3 she will think that someone isn’t wearing a hat when they are. If 10% of the Edinburgh population wears a hat, what are the chances that the suspect was really wearing a hat?
Needless to say, most visitors to the exhibit found these questions very difficult, but that was the point. We wanted to teach people not to trust their intuition when it comes to probability, and especially not if they are in a jury on a court case!
Many thanks to all who visited us in the Museum and played all our games with us! I hope you all had a good time and learnt something new.