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# Why know any algebra?

A recent meme circulating on the internet mocked a US government programme (ObamaCare) saying that its introduction cost \$360 million when there were only 317 million people in the entire country. It then posed the rhetorical question: “Why not just give everyone a million dollars instead?”

It may have been a joke but some nevertheless found this argument compelling. Their mistake was quickly pointed out  to give everyone a million dollars would cost 317 million million dollars but some persisted with the error. Their reasoning seemed to go “if you have 360 dollars you can give 317 people one dollar each so if you have 360 million dollars you can give 317 million people a million dollars each!” Others explained: “No Joe, just imagine you have 360 boxes, each containing a million dollars cash. When you go to give each person one of them you will run out after 360 people, leaving the remaining 316,999,640 people empty handed.”

Everyone would agree that adults need to have a grasp of numbers to the extent that they can spot nonsense arguments like this. At the same time however, it may be said that the x and y stuff can safely be forgotten once you leave school as it is practically never used. Anyone will forget the details of any subject if they never go back to it. That is why occasional reading about things mathematical renews confidence and allows you to question what is going on when a topic becomes complex.

For instance, it will give you the power to ask killer questions in pretentious presentations. Often a couple of graphs and equations flashed up on the screen will cower an audience into submissive silence. Never put up with that. Ask the presenter how that equation relates to the topic of their talk. Better still, ask what each of the symbols in the slide stand for. You don’t need to know anything in particular about maths to do that and everyone will soon see how competent your presenter is.

Taking this a little further, it is genuinely useful to know some algebra as it lets you deal with simple mathematical problems and to know that you have them right. And this does happen in real life. A friend once gave a presentation pitch for a contract that involved two factors whose graph was a straight line. He laboured to explain this and an audience member lost patience and pointed out that the two quantities had an obvious linear relationship so of course the graph had to be a perfect straight line. My friend was made to look clueless and, not surprisingly, failed to land the contract. He explained to me later that what most annoyed him was that he had figured that out the night before, and had even written down the equation of the line and checked it was right. However, he lacked the nerve to say that in his presentation and so when it was pointed out to him, he looked stupid. With just a touch more algebraic confidence he could have carried the day.

It is a worthwhile skill just to be able to see an algebraic problem for what it is even if your own attempts to solve it are a bit clumsy. A recent example concerned the controversial film The Interview, which is about a fictional assassination plot of the North Korean leader, Kim Jong-un. A magazine article said that the film grossed \$15 million on the weekend of its release, that the cost of the movie was \$15 to buy and \$6 to rent, and that two million copies were distributed overall. The article went on to say however that the company did not state how many copies were rented and how many were bought. It seemed that it did not occur to anyone at the magazine that they ought to be able to figure that out. A person with some mathematical habits of mind however would at least pause to think, and then would get the answer somehow, as it is not difficult. Working in units of millions there are two equations here: r + s = 2 and 6r + 15s = 15; the first equation counts units of r (rentals) and s (sales) while the second equation counts the money. From these we may deduce that there were 1/3 million sales and 5/3 million rentals overall. (Google simultaneous equations for further details.)

“I have a dream, which is that people will not run for cover whenever anything mathematical appears but rather will pause, think a little, ask a question or two and, if still out of their depth, seek a more qualified person to clear the matter up.”

The most salutory experiences of harm caused by mathematical ignorance however often stem from probability questions, which can fool even intelligent and educated people. It is one thing not to be able to do a problem but it is quite another to imagine that you can do it and be seduced into an utterly false conclusion. As an example, the following question was put to a large group of medical students. A certain condition affects one person in 1,000 and a particular medical test will certainly give a positive result if the person tested has the disease but has a 5% probability of coming out positive for people who have not got the condition. A randomly chosen person tests positive. What is the probability that they have the disease?

The most popular response was that the test was 95% accurate so the probability that it was right was 95%. I’m afraid that answer is not only wrong but represents a mistake on a par with poor Joe’s analysis of the cost of ObamaCare. The correct answer is at the other end of the scale: the chance that the person actually has the condition is less than 2%.

This is a tricky question and even a mathematically aware person might find it difficult to answer. However, I hope that the same person would instinctively be sceptical of the guess of 95% as that number takes no account of the prevalence of the condition in the population, which surely affects the answer. If the condition were very rare, then the chances of a false positive must be high. A quick way to see the right answer is to note that for every 1000 people in the general population, one person will have the condition but about 50 (5% of 1, 000) will be patients who generate a false positive; for that reason a random positive test only has a chance of one in fifty-one of detecting a person with the disease.

We might hope that qualified doctors would know how to interpret any test results that they call for. However, bad mistakes may still happen in serious situations such as court cases where an ‘expert’ witness makes a probability statement. Landmark cases involving cot deaths have led to gross miscarriages of justice. For example, once a probability statement on the likelihood of DNA matches is accepted as fact by the court, there may be only one verdict possible, and that may be the wrong one. I trust that lessons have been learnt from past errors but the risk of blunders remains unless any statement of probability is checked by a qualified statistician. Being an expert in the field of the testimony is not enough. Before a precise probability claim is admitted as evidence, it should be professionally scrutinised with the underlying assumptions, the calculation of the actual probability number and, just as importantly, its margin of error, all checked. I am not sure that would necessarily happen in a British law court.

In conclusion, I have a dream, which is that people will not run for cover whenever anything mathematical appears but rather will pause, think a little, ask a question or two and, if still out of their depth, seek a more qualified person to clear the matter up. It is a modest sounding dream but its realisation would make the world a better place.

Featured image credit: Maths and calculator. Public domain via Pixabay.

## Recent Comments

1. BeQui

Thank you for this article! It’s such a great illustration, I’ll be sharing it with my students!

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