Mathematics Probability

Precognitive dreams, Muggings and Deadly diseases – Problems by John Allen Paulos

I think you will agree with me when I say: Mathematics is hard.

For every handful of geeks out there, there is a huge majority that totally abhors the subject. I mean, come on, who loves mathematics?

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Enter John Allen Paulos. He bears an uncanny resemblance with the axe-wielding psycho ‘the Ripper’ from Last Action Hero.

Paulos has a problem (thank God, he doesn’t wield an axe). He thinks we do not give maths the importance it deserves. He has a name for the numbers-haters, too. ‘Innumerates’ is what Paulos calls them.

Over the years, Paulos has pushed the case of mathematics to make it more acceptable and more fun. And, he has garnered a fair share of attention, too, with his nifty little books. He throws some of the most intriguing, day-to-day life problems at his readers. These problems without the application of basic arithmetic and rules of probability can lead to wrong and sometimes horrifying conclusions.

In his book ‘Innumeracy‘, John Allen Paulos recounts several examples to demystify the thick veil around mathematics. I thought of sharing some of these brain-racking yet titillating problems with you. These problems are surely going to pique your interest in mathematics.

So here are three of my favorite mathematical problems from John Allen Paulos’ book:

1. Have our dreams got anything to do with reality? Do human-beings possess the power of precognition?

(Page 52, Innumeracy, John Allen Paulos, Penguin India)

Paulos assumes that 1 out of 10, 000 dreams matches the real-life situation of our lives. This implies that the chances of a non-match or a non-predictive dream are 9999 out of 10, 000.

Now let’s assume that each non-match day is an independent event. Thus, the probability of having ‘two’ successive non-matching dreams would be (.9999)*(.9999). In the similar vein, the probability of having N straight nights of non-match dreams would be (.9999)^N. For a non-leap year, the probability is (.9999)^365.

Since (.9999)365=0.964, we can conclude that 96% of people who dream every night would have non-match dreams whereas 3.6% will have predictive dreams (which they may think of as precognition). 3.6% translates into millions of precognitive dreams every year.

Hence, the occurrence of predictive dreams is no big deal, what does need explaining, however, is the non-occurrence of such dreams.

2. Should you be depressed when your doctor informs you that you have tested positive for a deadly disease?

(John Allen Paulos, p.62, Innumeracy, Penguin India)

Assume that there is a test for cancer which is 98% accurate. Assume further that only 1 out of 200 people actually have cancer, i.e. 0.5%. Now how would you react if your doctor breaks to you that you have tested positive for cancer? Crestfallen? Perhaps, yes, but Paulos insists you should be cautiously optimistic. Here’s how:

Imagine that of 10,000 administered tests for cancer, only 50 actually have cancer (0.5%). Since 98% of tests are positive, we will have 49 positive tests. Of the 9950 cancerless people, 2% of them will test positive, i.e. 199 of them. So, we have a total of 248 (49+199=248) positive tests and most of them are false positives.

Finally, the conditional probability of having cancer given that one tests positive, is only 49/248, i.e., 20%.

3. A man is mugged in downtown, New York and he claims that the mugger was a black man. However, when the crime scene is reenacted, it turns out the victim can only correctly identify the mugger about 80% of the time. What is the probability that his attacker was indeed black?

(Page 122, Innumeracy, John Allen Paulos, Penguin India)

Many people would jump to conclusion that the probability is 80% but the correct answer is again way off the mark. Paulos, in order to crack the problem, makes certain fair assumptions.

His assumptions are that approximately 90% of the population is white and only 10% black. Also, neither race is more likely to mug people and the victim is equally likely to misidentify the miscreant in both directions, white for black and black for white.

In view of these assumptions, out of 100 muggings, the victim would on average identify (80%*10)+(20%*90)=26 of the muggers as black. Out of these 26, only 8 are actually black men whereas he has mistaken 18 white males for black. Thus, the probability that the victim actually was indeed mugged by a black man is only 8/26, or approximately 31%.

Paulos demonstrates through this example that misinterpreting fractions could be a matter of life and death at times. I hope these problems manage to rekindle some of the lost romance towards mathematics.

Do check out my book review of Paulos’ book ‘Innumeracy’ here.

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