Writing grants and teaching, not to mention trying to get some actual research done, has taken up a considerable amount of my time this quarter. I mean, sheesh, I've barely had any time to read! This has, of course, made me grumpy. So when the publisher of The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow offered me a review copy of the book, I was very happy. I mean, I love probability and I love, um, well....you know :)
First of all, let me say that Mlodinow's book is preaching to the converted! A large portion of his book is devoted to showing how randomness is ubiquitous yet unrecognized and how we totally flub up understanding the role of randomness which surrounds us. Ah, these are good friends of mine! I believe both that randomness plays a greatly under appreciated role in our lives and that we humans are absolutely horrible at understanding the basics of probability (okay I'll admit, having taught probability, that even I have to scratch my head more thinking about probability, than say, doing a classical mechanics problem. Our brain is probably wired to incorrectly deal with probabilities due to some predator chasing us up a tree. Why mine seems to be better at classical mechanics is surely, then, a mystery.) Mlodinow's book is very well written, and passed the "didn't irritate me" hurdle I commonly encounter when reading popular science. Its full of lots of good stories about probabilities and even made me dream about probability while I was falling asleep reading the book. The Uncertain Chad expresses similar sentiments.
Now back to that first effect, that randomness is under appreciated. This is one of the reasons I have a hard time reading the type of blog article that says "how to succeed at problem X!" where X can be "getting into grad school," "landing a job in academia," "getting tenure," or any of the other many minutia that seem to waste large chunks of the academy's collective neuron trust. Sure such advice is usually good advice, but it amazes me how little those who have succeeding in, say, landing a faculty position fail to realize that it might be possible that they got there by some amount of chance. And these said offending blog articles are full of the type of recipedic (ooh, new word!) advice which completely and totally ignores the possibility that chance might be involved.
Okay, so I know: that's blasphemy! Of course all faculty members (they of the ivory cars and ivory teeth and ivory thoughts) got their jobs because of all their hard work and dedication and sheer brilliance. And even if there was some randomness involved, those other factors surely overwhelm chance, right? Well its exactly that last statement which I find so fascinating: actually what is occurring, I think, is much like what happens in the game of tennis.
Which brings me to one of the topics about probability, not discussed by Mlodinow in his book, but which struck me at an early age through reading Game, Set and Math: Enigmas and Conundrums by Ian Stewart. Suppose there are two tennis players who each have a fixed probability of winning a rally when they serve. For instance, suppose that player one, call him Dr. Bacon, wins on serve fourty percent of the time. Suppose that the second player, call him Dr. Stuffyfacultymember, wins on serve fifty percent of the time. What do you think the probability is that Dr. Bacon wins a match of three out of five sets? The answer, it turns out is about 4.9 percent. With such close probabilities of each player winning on their serve, the game of Tennis is structured in such a way that this translates into a lopsided probability of one player winning because repeated successes are necessary to win games, repeated games are needed to win sets, and repeated sets are needed to win a match. Ever since I saw this result growing up I remember being wary of those who've succeeded in games that aren't one-up events. While they may think that they are vastly superior to their competitive peers, in reality they need only be slightly better to show vastly different amounts of success.
Of course this doesn't invalidate the advice that is given: if small differences in upping your probability of succeeding at repeated trialled events can lead to long term wins, then you should definitely follow the advice. But, just remember, the advice givers are drunken tennis players whose delusions of prowess may just be an illusion of some basic probability.
I just read an excerpt from that book that appeared in the print edition of The Week two weeks ago or so. The excerpt dealt with the Monte Hall problem which is a problem I have, upon occasion, used in my classes.
For those who have no idea what I'm talking about (he's spewing bloody gibberish again!), Monte Hall had one contest on his game show that had three doors. Behind one was a new car and behind the other two were goats. The contestant gets to pick a door. Monte then opens one of the other two doors, always revealing a goat. He then offers the contestant the option of switching their choice of door from what they originally chose, to the only other unopened door left. The question is, is it better to switch?
This question was asked in the Ask Marilyn column of Parade magazine one year (she supposedly holds the Guinness Book of World Records record for highest IQ) and Marilyn gave the correct answer. The legendary and enigmatic mathematician Paul Erdös refused to believe this answer was correct until someone showed him a bunch of tapes from the show to confirm the statistical distribution. This is, of course, supposed to show, as The Pontiff mentioned above, that our brains aren't properly wired for probability.
So, are real probabilities Bayesian or frequentist? Hmmm... I smell a debate.
I couldn't agree more. I have always believed that the successful rationalize out the components of luck in their past.
This connects nicely to Grrlscientist's discussion of the role of contingency in evolution as shown by the evolution of citrate metabolism in ne lineage of e. coli.
Marilyn Vos Savant (no relation to me) can't possibly have the highest intelligence in the world, compared to Paul Erdos or anyone else of the Feynman, Gell-Mann, Hawking level of genius.
But good at self-promotion? Definitely. As with Mick Jagger or Madonna, I'll give her credit for being energetic and laughing all the way to the bank.
What happens to the tennis calculation if Dr. Bacon only wins 40 percent of his serves if it's not returned, 42% if a rally goes to two shots, 60% for longer rallies, Dr. Stuffyfacultymember has opposite numbers, and the lengths of the rallies has probability distribution p(n)? How does making the analogy of the tennis calculation with academic tenure work then? It seems problematic (disclosure: I don't have an academic job, my wife has tenure).
If you can call Dr. Stuffyfacultymember that, it seems likely that s/he had some form of early advantage, so that s/he won a three set match against Dr. Bacon, but has not won Wimbledon -- and it seems very unlikely to everybody else that they will win, but they still claim that they will win, next year, this year they had too much teaching.
In another wild extension of your argument, I'm not sure that I can say that Feynman was 2% smarter than I am if over a 40 year career he got ahead of me. How do we figure such odds, and what would they mean, anyway? Was he better than me at scrabble, and does that particular data point matter or not? There are a thousand other dimensions to ask about, most irrelevant, of course, but only after the event do we know them to be so. Calculations of probability are only legitimate (worthwhile, ...) if there are restricted formal rules to a game, which tennis only has if we make moderately gross assumptions, and academia surely does not have (unless you have a good mathematical model for Mind that you haven't told anyone about). We often simulate or model complex situations probabilistically to good success, but there is a point somewhere beyond tossing coins, dice, and cards where it remains a pragmatic question whether a particular statistical model will be empirically effective, and the question of causality generally remains obscure even if a statistical model works. Sports results predictions from even quite extensive statistical datasets are not that good, right?
Well, I don't hold a lot of stock in IQ tests anyway, but, on the other hand, she got it right (the Monte Hall problem) and Erdös got it wrong - and refused to believe it until he saw the tapes (at least according to the Mlodinow book).
I never understood why the Monty Hall problem is not intuitive. The fact that you have a 66% chance of picking the booby prize on the first shot means you've probably got what you don't want so of course you want to swap. Even more so for Deal or No Deal. One out of 26 is not good, so you probably are not holding the big number case at the end and of course swapping is a good idea.