The arrival of the latest addition to the Royal Family got me thinking about a probability problem which is quite straightforward on the face of it but which most people get wrong:
A pregnant woman attends for her routine ultrasound scan, and the sonographer diagnoses twins, telling her that one of them is certainly a boy, but they didn’t get a good enough view of the other to be able to tell the sex. When the mother is subsequently delivered, what is the probability that the first-born twin will be male?
Since this is a teaser in mathematics and not obstetrics you should assume that these are fraternal rather than monozygotic twins (i.e. they are not identical), as allowing the possibility of identical twins automatically increases the chance that they are both of the same sex. Furthermore, for the purpose of the exercise I want you to assume that the birthrate of boys and girls is the same. We know that in reality it isn’t. John Arbuthnot demonstrated in his 1710 paper An Argument for Divine Providence that the human sex ratio is roughly 21 boys born for every 20 girls (he derived this from Christening records in London between 1629 and 1710). Nevertheless I prefer to keep the arithmetic simple in this case.
Of course its safe. I’ve done this dozens of ti…
Probability is something that we all have an instinctive feel for. In so many of the decisions we make it is important to have a rough idea of the chances of it going one way or another, and most people are quite confident in their assessment of risk. Unfortunately they are usually wrong, and probability is one area where our instincts and gut feelings completely mislead us.
How many people do you know who are nervous of flying, but would be quite happy to make the journey on the M25 between Heathrow and Gatwick airports? The risk of dying in an air crash is about 1 per million flights taken, compared with 1,800 road deaths in the UK per year (and about 27,000 seriously injured), just under half of these being occupants of cars. About 33 million people in the UK hold full driving licences so a back-of-the envelope calculation suggests that something like 1 in 40,000 drivers and their passengers are killed each year. Flying is clearly much safer.
I don’t suppose many people take the trouble to calculate these risks, but you might have thought that regularly witnessing accidents might give us pause. While I was working I would usually see two accidents every day on the M25 during my commute between Chiswick and Guildford, with associated traffic jams due to the morbid curiosity of rubbernecking passersby. Nevertheless, there is always a slight anxiety manifesting, if not a downright phobia even, on boarding a plane which is absent when getting into a car.
Part of this is familiarity, as it is hard to keep in mind that things we do every day can be dangerous. This is an insidious risk, since they only need to go wrong once…
Part of it is the sense of control. When you fly, you have to hand over the responsibility for your safe arrival to the pilot and his crew, but when you drive, then you are in control of what happens. After all, you know what a good driver you are, much better than average (most motorists believe this, even though half of all drivers are worse than average by definition), more than good enough to make up for the other bad drivers on the roads. Yes indeed!
My grandmother smoked all her life and she lived to be 89
Another example is cigarette smoking. In 1956 Doll and Peto published definitive evidence that smoking increased the risk of lung cancer, and it has been common knowledge since then, with more and more dangers revealed by further studies. The recent fall in smoking has been largely due to legislation and the rise of vaping rather than people’s concerns about their health. On average smokers die 7 years younger than their non-smoking peers and experience an additional 12 years of poor health. But we tend to discount these high risks since the decline in health is gradual and doesn’t start immediately.
Actually my grandmother did smoke, black Sobranie cigarettes when she could get them, otherwise my grandfather’s Rothmans. She didn’t like them to touch her lips, however, so she kept a bowl of rose petals to wrap around the tip when she smoked them. She had long white hair and long strings of pearls and liked to wear turquoise dresses. She would blow smoke rings to entertain us when we were small. She lived to be 89 and died of something unrelated to smoking. However you load the dice, they don’t always fall the way you expect them to.
Maybe, just maybe…
And what about the Lottery? The chance of winning the jackpot in the UK Lottery is roughly 1 in 14 million, compared to the chance of being murdered, which is about 1 in 83,000. Of course there are lesser prizes which are given out more frequently, but less than half of the price of the tickets goes towards the prizes, the rest going in administrative costs and supporting charities.
So we know that most people are hopeless when it comes to probability, and this manifests in the bad life choices they make. Lets have a look at some more simple problems and see how we do.
Two coins in a fountain…
Imagine that I have tossed two coins and then put my hands over them once they have landed on the table. You are tasked with working out how they have fallen. Note that this is slightly different from the twins problem earlier, where you are trying to predict what will happen when they are born. This time the coins are already one way up or another, you just don’t know the result.
I’ll start with a trivial problem:
I have looked at the coins and they are both heads. What is the probability that they are both heads?
Obviously the answer is one. Since I have already given you complete information about how they landed you can say this with certainty.
I have looked at the coin in my left hand, and it is heads. What is the probability that they are both heads?
The difference here is that I have given you less information, and so you can’t be as certain about the outcome. I hope you don’t find it too difficult to work out that the answer is 0.5, or 1/2 (or 1 in 2,or 50:50 or odds of 1 to 1, which are all different ways of saying the same thing).
I haven’t looked at either coin. What is the probability that they are both heads?
This time I haven’t given you any information, so you will be less certain still that they have both come up heads. Because you have no additional information about the two coins, the probability is the same as the probability of them both landing heads in the first place. Each coin is equally likely to come down heads or tails, and they are independent of each other, so there are four equally possible outcomes: HH, HT, TH, TT. In only one of these cases are there two heads, so the probability is 1/4 (or 1 in 4 or 0.25).
I have looked at one of the coins and it is heads. What is the probability that they are both heads?
On the face of it that sounds rather like the second problem, but in fact I have given you less information since I haven’t specified which coin I looked at. You may be starting to realise that the probability of both coins being heads is really a measure of our uncertainty over whether they are both heads, after all, they have both already been tossed, and somebody else with different information is going to come to a different answer. Working on that basis you might eventually conclude that the answer is somewhere in between 1 in 4 (knowing nothing) and 1 in 2 (knowing that a particular coin is heads and only being uncertain about the other one). If that is how you are thinking then you are right. An easy way of looking at it is that your information allows you to rule out the possibility that the coins both landed on tails, leaving you with three equally likely arrangements, HT, TH and HH. The answer is therefore 1 in 3 (or 1/3, 0.333…).
Perhaps you are still nursing a nagging feeling that I have been cheating somewhere. This is because our intuitions are so wrong when it comes to probability that the only way to get it right is to work it out and accept the answer even if it feels wrong. But I hope you are starting to appreciate that there is a difference between uncertainty over what is going to happen (aleatoric uncertainty) and uncertainty over what has happened already (epistemic uncertainty). The first person to realise this, and to come up with mathematical approaches to deal with incomplete information, was the Reverend Thomas Bayes, an 18th-century Presbyterian minister who lived in Tunbridge Wells.
Back to the twins
Coming back to the problem about the twins, mathematically it is very similar to our fourth problem with the coins, since we know that at least one twin is male but we don’t know which one. The options are therefore BG, GB and BB as we have ruled out GG. Two out of these three equally likely possibilities have a male first-born twin, so the answer is 2 in 3, 2/3 or 0.6667…
I will leave you with a problem that was once introduced at the start of a conference attended by professional mathematicians and statisticians, who could not agree on the answer, although it is not that hard to work it out using the principles I have already outlined. However, even people who have studied statistics often know less about probability than they think (along with the rest of us). Generally speaking, however, they are more pig-headed when considering that they might be mistaken.
I have two grandchildren, one of whom, born on a Tuesday, is a girl. What is the probability that they are both girls?
Hint: I have given you more information than if I hadn’t said anything about the weekday the girl was born on, but less information than if I had specified whether she was the elder or younger child. I will let you have the answer in a future post, once I have had a few suggestions in. If I am feeling wicked I might even bring up the story of Marilyn and the Goats. Once again, please assume that babies are equally likely to be born on any weekday (which is not actually the case) and that there are as many boys born as girls.
Did you know that in Ghana children are named according to which day of the week they were born?