We have come to the third week of semester 1 in Academic Year 2016/2017! It can be a little difficult getting back to full-time studying after the summer holidays, and we would like to offer some of you a breather from your lectures and tutorials. Camille Mau, a second year CN Yang scholar with a passion for mathematical sciences has prepared a piece on The Strangeness of Probability – Bayes’ Theorem. Without further ado, let’s take a look at
The Strangeness of Probability – Bayes’ Theorem
BY CAMILLE MAU
I claim: 50% of people will be influenced by this article.
Before you write me off as a pretentious prat, hear me out. Suppose you were faced with the following question:
“A new test for Disease X came out the other day. It is free for all, so you go for it. The test is known to give false positives and false negatives at times. A person who has Disease X will test negative 1 in 100 times, and a person who does not have Disease X will test positive 1 in 100 times. 1 in 10000 people have Disease X. You test positive. How worried should you be?”
The question posed above is a variant of a common question on the Internet which tends to trick many. Human nature instinctively gravitates toward only the numbers that we see, and can be tricked to deviate from the actual math.
In this scenario, you have been hit with the very low chance of having Disease X, and will most likely test positive for it. So what’s the problem here?
The problem is that the test is not at all reliable. Let’s do some number-crunching. First let’s draw a table.
From this table we can see that about 1 in 100 people who test positive have Disease X. In other words, there is only a 1% chance which you have Disease X! (For the mathematically-anal, the actual number is 1 in 102.)
What am I trying to show with this example? It is that probability is not defined by just the numbers you see and interpret in your head. While the information is there, we all have a preconceived notion of facts and figures in our heads. These ideas may be used to trick you.
Let’s discuss another well-known example. You have flipped a fair coin ten times, and the results have all been heads. Instinctively, one would answer “tails” when asked to predict the next flip. After all, it is very improbable that a fair coin flips heads for 11 times in a row.
This, too, is fallacious thinking. It is true that it is very improbable that a coin would flip heads 11 times in a row. It is also true that you have already flipped the coin 10 times already. The question at hand is not asking about the probability of flipping 11 times in a row. It is asking for the probability of flipping heads, given that you have already flipped 10 heads in a row. The difference here is that the past events have no bearing on what you are going to flip next. You want the probability of flipping 11 heads, given that you have already flipped 10 heads in a row prior. In other words, this is a conditional probability question.
(A conditional probability is the probability of one thing (A) happening given that another thing (B) has already happened. We write this as P(A|B).)
Now, let’s introduce the simple statement of Bayes’ Theorem. Don’t worry, I won’t ask you to do any number-crunching.
In words, the probability of an event A happening given that some event B has already happened, is equal to the multiplication of the conditional probability of observing event B happening given that event A has happened and the probability that event A has indeed happened, divided by the probability of B happening.
Perhaps it would be easier to visualize the numerator as the probability of B happening and A happening simultaneously, over the probability of B happening. In the context of the coin flip, we can take A = “the coin flips heads 11 times” and B = “the coin flips heads 10 times”. Then,
so it doesn’t matter if you choose heads or tails, the probability of both is the same due to the fact that past events are past. Similarly, in the case of the flawed Disease X test, we just want the probability that you actually do have Disease X given that you test positive, considering the probabilities that you indeed do have Disease X in the real world.
In the end, you may ask: Isn’t Bayes’ Theorem just something we might have learned in high school? What’s so special about it?
The answers are: Yes, and nothing. The theorem is not some miraculous mathematical statement on the verge of breaking the universe. Unlike the Riemann-Zeta Hypothesis, or Quantum Physics, or the number 42.
What is more important are the insights that the theorem gives us about the way the world can be structured to trick you. We return once more to the Disease X example. A test which gives false results only 1 time in 100 trials can be marketed as 99% accurate. After all, it does give accurate results 99% of the time. There is no conflict here, but we are hard-wired to see this 99% result and believe it applies to every related probability instantly. If we test positive, we think that there is a 99% chance we have the illness, and so on. We have just seen that that form of thinking is no more than a flawed train of thought.
I am not saying not to trust what companies and products say about their statistics. Whatever is reported, barring corruption, is accurate. There is no question about how accurate a test indeed is, or the percentage values reported, for instance. Accept those values, and then consider what it really means. What does the probability really claim? Is the claim universally and truly accurate, or is there some “fudge factor” necessitating further thought? Once you have mastered the art of not getting tricked, you will have a brand new worldview, one which lets you make decisions better.
So, my question to you now: At the start of the article, I made a very bold claim. And therein lies the question: What does my claim really say?
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Featured image: Mathematics [Digital image]. (n.d.). Retrieved from https://admissions.carleton.ca/degrees/mathematics/