Sunday, April 25, 2021

269: A Good Use of Downtime

 Audio Link

The recent observation of Holocaust Remembrance Day reminded me of one of the stranger stories to come out of World War II.    John Kerrich was a South African mathematician who made the ill-fated decision to visit some family members in Denmark in 1940 just before the Germans invaded, and soon found himself interned in a prison camp.    We should point out that he was one of the luckier ones, as the Germans allowed the Danes to run their prison camps locally, and thus lived in extremely humane conditions compared to the majority of prisoners in German-occupied territories,   But being imprisoned still left him with many hours of time to fill over the course of the war.     Kerrich decided to fill this time by doing some experiments to demonstrate the laws of probability.


Kerrich’s main experiment was a very simple one:   he and a fellow prisoner, Eric Chirstensen, flipped a coin ten thousand times and recorded the results.    Now you might scratch your head in confusion when first hearing this— why would someone bother with such an experiment, when it’s so easy for anyone to do at home?    We need to keep in mind that back in 1940, the idea that everyone would have a computer at home (or, as we now do, in their pocket) that they could use for seemingly endless numbers of simulated coin flips, would have seemed like a crazy sci-fi fantasy.    Back then, most people had to manually engage in a physical coin flip or roll a die to generate a random number, a very tedious process.      Technically there were some advanced computers under development at the time that could do the simulation if programmed, but these were being run under highly classified conditions by major government entities.   So recording the value of ten thousand coin flips actually did seem like a useful contribution to math and science at the time.


So, what did Kerrich accomplish with his coin flips?    The main purpose was to demonstrate the Law of Large Numbers.  This is the theorem that says that if you perform an experiment a large number of times, the average result will asymptotically approach the expected value.    In other words, if you have a coin that has 50-50 odds of coming up heads or tails, if you perform lots of trials, you will over time get closer and closer to 50% heads and 50% tails.   Kerrich’s coins got precisely 5,067 heads, and over the course of the experiment got closer and closer to the 50-50 ratio, thus providing reasonable evidence for the Law.    (In any 10000 flips, there is about an 18% chance that we will be off by at least this amount from the precise 50-50 ratio, so this result is reasonable for a single trial.)   


Of course, it might make sense to request another trial of 10000 flips to confirm, for improved confidence in the result.   But apparently even in prison you don’t get bored enough for that— in his book, Kerrich wrote, “A way of answering the… question would be for the original experimenter to obtain a second sequence of 10000 spins,   Now it takes a long time to spin a coin 10000 times, and the man who did it objects strenuously to having to take the trouble of preparing further sets.”


Kerrich and Christensen also did some other experiments along similar lines.    By constructing a fake coin with wood and lead, they created a biased coin to flip, and over the course of many flips demonstrated 70/30 odds for the two sides.   This experiment was probably less interesting because, unlike a standard coin, there likely wasn’t a good way to estimate its expected probabilities before the flips.   A more interesting experiment was the demonstration of Bayes’ Theorem using colored ping-pong balls in a box.   This theorem, as you may recall, helps us calculate the probability of an event when you have some knowledge prior conditions that affect the likelihood of each outcome.    The simple coin flip experiment seems to be the one that has resonated the most with reporters on the Internet though, perhaps because it’s the easiest to understand for anyone without much math background.


In 1946, after the end of the war, Kerrich published his book, “An Experimental Introduction to the Theory of Probability.”.    Again, while it may seem silly these days to worry about publishing experimental confirmation of something so easy to simulate, and which has been theoretically proven on paper with very high confidence anyway, this really did seem like a useful contribution in the days before widespread computers.    The book seems intended for college math students seeking an introduction to probability, and in it Kerrich goes over many basics of the field as demonstrated by his simple experiments.    If you’re curious about the details and the graphs of Kerrich’s results, you can read the book online at openlibrary.org, or click the link in our show notes at mathmutation.com .   Overall, we have to give credit to Kerrich for managing to do something mathematically useful during his World War II imprisonment.    


And this has been your math mutation for today.




References:  


Saturday, March 20, 2021

268: The Right Way to Gamble

Audio Link

At some point, you’ve probably heard an urban legend like this:   someone walks into a Las Vegas casino with his life savings, converts it into chips, and bets it all on one spin of the roulette wheel.    In the version where he wins, he walks out very wealthy.   But, in the more likely scenario where he loses, he leaves totally ruined.    This isn’t just an urban legend, by the way, but it has actually happened numerous times— for example, in 2004, someone named Ashley Revell did this with $135,000, and ended up doubling his money in one spin at roulette.    I most recently read about this incident in a book called “Chancing it:  The Laws of Chance and How They Can Work for You”, by Robert Matthews.   And Matthews makes the intriguing point that if you are desperate to significantly increase your wealth ASAP, and want to maximize the chance of this happening, what Revell did might not be entirely irrational.


Now, before we get into the details, I want to make it clear that I’m not recommending casinos or condoning gambling.    Occasionally while on vacation my wife & I will visit a casino, and here’s my foolproof winning strategy.    First you walk in, let yourself take in the dazzling atmosphere of the flashing lights, sounds, and excitement.    Then walk over to the bar, plop down ten bucks or so, and buy yourself a tasty drink.   Sit down at the counter, take out your smartphone, start up the Kindle app, and read a good book.   (The Matthews book might be a nice choice, linked in the show notes at mathmutation.com.)    Then relax in the comfortable knowledge that you’re ahead of the casino by one pina colada, which you probably value more than the ten dollars at that particular time.   Don’t waste time attempting any of the actual casino games, which always have odds that fundamentally are designed to make you you lose your money.


Anyway, getting back to the Revell story, let’s think for a minute about those rigged odds in a casino.   Basically, the expected value of your winnings, or the probability of winning each value times the amount of money, is always negative.   So, for example, let’s look at betting red or black in roulette.  This might seem like a low-risk bet, since there are two colors, and the payout is 1:1.   When you look closely at the wheel, though, you’ll see that in addition to the 36 red or black numbers, there are 2 others, a green 0 and 00.   Thus, if you bet on red, your chances of winning aren’t 18/36, but 18/38, or about 47.37%.   That’s the sneaky way the casinos get their edge in this case.   As a result, your expected winnings for each dollar you bet are around negative 5.26 cents.  This means that if you play for a large number of games, you will probably suffer a net loss of a little over 5% of your money.


So let’s assume that Revell desperately needed to double his money in one day— perhaps his stockbroker had told him that if he didn’t have $200,000 by midnight, he would lose the chance to invest in the Math Mutation IPO, and he couldn’t bear the thought of missing out on such a cultural milestone.   Would it make more sense for him to divide his money into small bets, say $1000, and play roulette 135 times, or gamble it all at once?   Well, we know that betting it all at once gave about a 47% chance of doubling it— pretty good, almost 50-50 odds, even though the casino still has its slight edge.   But if he had bet it slowly over 100+ games, then the chances would be very high that his overall net winnings would be close to the expected value— so he would expect to lose about 5% of his money, even if he put his winnings in a separate pocket rather than gambling them away.    In other words, in order to quickly double his money in a casino, Revell’s single bold bet really was the most rational way to do it.


And this has been your math mutation for today.




References:  

https://www.amazon.com/dp/B014RT1M1U/








Sunday, February 7, 2021

267: Free Will Isn't Free

 Audio Link

If you’ve browsed the web sometime in the last three decades, especially if you visit any New Age or philosophy websites, you’ve probably come across the argument that quantum mechanics proves the existence of free will.   Superficially, this seems somewhat plausible, in that quantum mechanics blows away our traditional idea that we can fully understand the future behavior of the universe based on observable properties of its particles and forces.   Free will seems like a nice way to fill in the gap.   But when you think about it in slightly more detail, there seems to be a fatal flaw in this argument.   So let’s take a closer look.


To start with, what is so special about quantum mechanics?   This is the area of physics, developed in the 20th century, that tries to explain the behaviors we observe in subatomic particles.   What’s bizarre about it is that according to its calculations, as Einstein stated, God is playing dice with the universe:  we can calculate probabilities of the position and momentum of particles, but not the exact values until we observe them.     For example, suppose I am throwing a toy mouse for my cat to catch.  Since this is a macroscopic object, classical physics works fine:  my cat can take out a calculator, and based on the force I throw with, the mass of the mouse, and the effects of gravity, he will be able to figure out exactly where to pounce.     But now suppose I am throwing a photon for him to catch.   Even if he knows everything that is theoretically knowable about my throw, he will not be able to calculate exactly where it will land— he can only calculate a set of probabilities, and then figure out where the photon went by observing it.    Of course, my cat uses the observation technique even with macroscopic mice, so no doubt he has read a few books on quantum physics and is trying to use the most generally applicable hunting method.


But how does this lead to free will?   The most common argument is that since things happen in the universe that cannot be precisely calculated from all the known properties of its particles and energies, there must be another factor that determines what is happening.   Many physicists seem to believe that a hidden physical factor has been largely ruled out.   That means the missing piece, according to the free will argument, is likely to be human consciousness.    When the activities of the particles in our brain could determine multiple possible outcomes, it is our consciousness that chooses which one will actually happen.    The quantum events in my brain, for instance, could lead with equal probability to me recording a podcast this afternoon, or playing video games.   My free will is needed to make the final choice.


Now time for the fundamental flaw in this argument.    Suppose the quantum activities in my brain do ultimately give me a 50% chance of recording a podcast this afternoon, and a 50% chance of playing videogames.   The implicit assumption in the free will argument is that if the properties of my brain determined completely that I would make a podcast this afternoon, then I would not have free will.   Since there are two alternatives and I have to choose one, there is free will.    However, remember that the quantum calculations provided an exact probability for each event, not just a vague uncertainty, and these probabilities have been well-confirmed in the lab.   If I’m required to roll a 6-sided die, and record the podcast if numbers 1-3 come up, or play videogames if 4-6 comes up, aren’t I just as constrained as if I were only allowed one of those options?   I still don’t get any freedom here.    Either way, it’s not a question of my consciousness, it’s just another calculation, though one whose result cannot be predicted.   If I’m forced to do certain things with known probabilities, that’s the opposite of free will.


There is one more wrinkle here though, which may rescue the free will argument.   Suppose we interpret the quantum calculation slightly differently.   Yes, there is a 50% chance of me recording the podcast, and 50% of playing a videogame.   But maybe this isn’t the universe rolling a die— maybe it’s a measure of the type of personality created by the sum total of quantum interactions in my brain.   So rather than being forced to roll a virtual die and decide, the universe has just built me into the kind of guy who, given a choice, would have a 50-50 chance of podcasting or gaming this afternoon.    All the quantum calculations about my brain activity are just figuring out the type of personality that has been created by its construction.   That argument rescues free will in the presence of quantum probabilities.   Though it does put us in the odd position of wanting to ascribe some kind of conscious will to subatomic particles observed in a physics lab, or to assume some hidden spirits in the room are making decisions for them.


So what’s the answer here?    Well, I’m afraid that as often happens with these types of questions, we will not resolve it in a 5-minute podcast.    You can also find some more subtle arguments that incorporate other aspects of quantum physics, though those don’t look too convincing to me either.   If humanity truly doesn’t have free will, I’ll look forward to the day when someone writes a treatise on how a large Big Bang of hydrogen atoms fundamentally leads to Math Mutation podcasts a few billion years later.


And this has been your math mutation for today.




References: