Recently, I came across two posts on probability, both of which possessing some serious issues. This is actually a common occurrence, since probability is one of those things that confuses people and is, in many ways, counterintuitive. What’s interesting about both of them is that the answers revealed are misleading in the same manner as the intuitive solutions.
The first is, “How unfathomable were your odds of coming into being?” This one is annoying in that there are countless ways that this question can be interpreted, and only one is displayed. Especially painful is the tagline at the top of the infographic which reads, “…and reveals that the odds of you existing are almost zero.” This is a classic case of misinterpretation – since you’re reading this, you exist, so the odds of you actually existing are 1 in 1, alternately listed as 1:1, 1/1, or simply 1, otherwise known as “guaranteed” (Jerry Coyne and PZ Myers get it right.) The question implied by the solution is more along the lines of, “What are the odds of known variable factors within common knowledge leading to exactly your point in time, rather than some other?”, which isn’t even accurate in itself. Start with, the odds of your dad meeting your mom being, as listed, 1 in 20,000 – that’s not true at all. You have to factor in how much your dad and mom might have traveled, how big the town they lived in was, how many people worked in the same place, and so on. And since a certain percentage of people marry their childhood sweetheart, you can run some figures on how many people were in the school they attended, within a few years of age, or in the immediate neighborhood. It’s way, way off base to create a random figure of available women (or men) and assume that anyone could have met any of them, like they were playing Spin The Bottle in a very large room…
The second part of this is, you can provide any particular scenario, compared against the number of variations, and marvel lightheadedly at the number involved. What are the chances, for instance, of the atoms within your body being at exactly such-and-such coordinates at some particular picosecond, compared to anyplace else in the universe? I could produce a number even greater than that displayed by the infographic, but again, this is meaningless; in another picosecond, the atoms have shifted position and attained yet another astoundingly high number, which turns your entire life into a series of events so improbable that you should cease existence almost immediately. Then again, it’s a fairly high probability that you will be someplace on the surface of the earth, wherever it is in the universe, and so that astoundingly high number drops drastically. In grim reality, the atoms within your body are very likely to remain in the immediate vicinity, within a fraction of a millimeter of where they were before. The variations that take place, for virtually any action or process, are usually quite small from moment to moment, often influenced by environmental factors. When your body moves at all, it moves to an area immediately adjacent to where it once was, often influenced by the trend of how you had been moving previously – you don’t reverse direction or shoot off at random angles. While your overall path can vary greatly from a starting point, how it gets there is through a series of tiny variations, many of which are extremely likely. Interestingly, this is a great analogy for evolution, which produces significant changes in small increments over long periods of time.
The likelihood of you being you, as in, thinking and behaving in a certain way, is actually much higher than implied by this whole situation, as well. You probably received your education at the nearest school, and from the same parents as any siblings, and from whatever situations you might have found yourself within (say, being shipped off to summer camp.) Those factors all serve to narrow down the chances of certain outcomes; if you were born in the US, for instance, the chances of you speaking nothing but Farsi are pretty slim. The chances of you attending a summer camp in Australia, also slim – the camp you attended is probably within easy driving distance, just due to common convenience. You probably look human, breathe air, eat protein, and shit shit. If you were truly as unique as implied by the quoted figure, you probably wouldn’t even have offspring yourself, because you wouldn’t be able to find a spouse that you had anything in common with ;-)
The other post is not as bad, and even better, the commenters are doing a good job of correcting it. Jason Rosenhouse at EvolutionBlog presents a chestnut that challenges some assumptions while falling blindly into others, making kind of a mess of the whole thing. As he mentions, there’s a probability puzzle called the Monty Hall Problem, where the actual probabilities are different from what intuition tells us it should be – mostly because “Monty” knows the goal that the contestant seeks. Very briefly: you choose one of three doors that might have a prize. Monty opens one of the two remaining that you did not choose, revealing no prize. Are your chances better with switching, or staying with your original choice? Look at it this way: you had a 2 in 3 chance of being wrong initially, and if so, Monty just showed you which one to pick. The chances are doubly in your favor by switching, as long as you don’t pick the door Monty already opened…
So, a man comes up to you on the street and says, “I have two children and one is a son born on a Tuesday.” What is the probability that the other child is also a son? Most people would say 1 in 2, or 50%.
Now it gets screwed up. Jason points out that,
It follows that the sexes of his two children, ordered from oldest to youngest, are either BB, BG or GB.
But this is wrong, because the other scenario, not listed, is also Boy/Boy, only this time the son named is the youngest and not the eldest. The oldest/youngest factor is a red herring. In other words, there are only three scenarios for two kids (both girls, both boys, and one of each) and one has been eliminated. It’s still 1 in 2.
Then, he goes on to speculate what affect Tuesday has on the situation, and the chances of the other child being born on whatever day of the week. This is called needlessly multiplying probabilities, because adding further options (hair color, time of day, and so on) does not change the original factors at all.
Then, he introduces the factor of how this particular man might have been selected. Now, we’ve ventured outside of the realm of simple word problems and into real-world scenarios that we have not been given information about. “What if the man was selected because he had one son born on a Tuesday?” Yes, you can start adding in contingencies, but all this does is show that probabilities can only be calculated within rigid circumstances. I can immediately ask, “Selected from where?” and find that this experiment was done in a town with more girls than boys in the population, and the probabilities are thus biased from the 1 in 2.
The problem, for instance, never allowed for the probability that one of the children was a hermaphrodite. We have to consider that the man might be insecure (probably over the concern that his child would turn out both gay and lesbian) and thus want to prove himself Mr. Clever Dick by putting one over on a total stranger, so the likelihood that his child is hermaphroditic is even higher than the normal percentage within any given population – it’s a great opportunity for such a trick question (though rather taxing on the child who is usually called upon to prove it.) And since you are either a dick or you’re not, the chances of the man being a dick are 1 in 2, right? Unless the man is a driver in North Carolina, in which case it’s 17 in 19…
[Okay, you’re going to love this. I paused and saved the draft right here, and checked my spam folder for its frequent new additions. There were two messages in there, one from a commenter with “shemale” in the name. What are the chances?!]
What all of this does serve to illustrate, however, is one way in which critical thinking can be applied, since probabilities are very frequently misused. And they can rarely be applied to a given situation with any degree of accuracy anyway, because environmental variables in real-world situations are too vast to calculate. I frequently remind people that math is an abstraction, and is only used with certain aspects being assumed or ‘given.’ Two oranges rarely ever equal twice the mass of one orange, and even the surface of the orange can only be calculated on a broad level, because at the atomic level there is little than can be measured…
This also puts paid to the claims, ever so frequently given, that the chances of such-and-such event (complex life, evolution, bacteria flagella…) occurring is a specific number, like one-in-a-billion. There is, literally, no way in which such a thing could ever be calculated, because there is no way to know all of the factors involved. Therefore, there is a 1 in 1 chance that any such quoted number has been pulled out of someone’s ass, and can safely be dismissed as a blatant lie intended to influence your thinking.
Glad to be of help.