Numeracy #17: Markov Chains

Suppose we want to predict the chance that it rains tomorrow using the variables available to us today. To make it simple, let’s say there are two variables that we are tracking: raining or sunny. If it is sunny today, the chance of it being sunny tomorrow are higher, and if it is raining today, the chance of it raining tomorrow are higher. This seems like a reasonable, though simplified, way to model reality. When tomorrow comes, we would just repeat the process.

Similarly, suppose we want to predict the next word that will be typed on a cell phone. We could look at every word that has ever been typed on that phone and find the most frequent one, or we could just look at the most recently typed word and see what word usually comes after it.

In both cases, we are using the most recent information available to us to predict the future and ignoring everything that came before. When the future comes, we transition and repeat the process. This way of modeling the world is known as a Markov Chain and it has some powerful applications.

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Numeracy #16: Blue and Red Marbles Puzzle

Question: You have 50 blue marbles, 50 red marbles, and two jars to put them in. A marble will be selected at random from a jar selected at random. How do you divide the marbles among the jars as to maximize the probability of choosing a blue marble. You must use all the marbles.

As always, try solving the problem yourself or keep reading for the solution.

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Numeracy #15: Male Population Puzzle

Question: Consider a country where every family wants to have a boy. Each family continues having babies until the arrival of a boy and then they stop reproducing. After years go by, what will be the proportion of boys to girls in the country? Assume that there is a 50% chance of having either a boy or a girl.

As always, try solving the problem yourself or keep reading for the solution.

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Numeracy #14: Probability of Car Passing By

Question: The probability of a car passing a certain intersection in a 30 minute window is 0.95. What is the probability of a car passing the intersection in a 10 minute window? Assume that the cars are randomly distributed.

Try solving the problem yourself or keep reading for the solution.

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Numeracy #13: The Ant Triangle Puzzle

Question: Three ants are sitting at the three corners of an equilateral triangle. Each ant randomly picks a direction and starts to move along the edge of the triangle. What is the probability that none of the ants collide?  What is the probability that n ants sitting at n corners wont collide?

Assume that the ants travel at the same speed and there’s no pheromone communication going on. Try to solve the problem yourself or keep reading for the solution.

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Numeracy #12: Rule of 72

Imagine that we just opened a new savings account that pays 10% interest compounded annually. In reality, we would get something closer to 0.1% interest compounded continuously, but let’s keep the numbers large and whole for now.

If we opened the account with $100, after a year we would $110, and after two years we would have $121. How long will it take us to double our money? You may have heard of the Rule of 72 which is a trick for answering this question. Before we get into how it works, let’s review how compound interest works.

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Numeracy #11: Statistical Significance

Imagine you just developed a new strategy for coin flipping. Right before you release the coin, you blink 4 times. You want to test if this strategy will result in more heads than tails, so you set up a test where you flip a coin 20 times. You observe 13 heads and 7 tails. Your strategy resulted in heads 65% of the time! You rush out to start gambling on coin flips.

You probably already realized the error in our analysis. Even if our new strategy had zero impact, we would still expect to flip 13 heads about 7.3% of the time. This raises an important question: how do we know if our results are the product of our strategy or randomness? We don’t, but we can set up our experiment in a way that the odds of a random result are so small, that we are confident the effect is from our strategy.

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Numeracy #10: Nash Equilibrium

Imagine two criminals, Bob and Sam, who were arrested today for selling drugs. It’s an open and shut case, but the district attorney has a hunch that Bob and Sam were involved in a murder that took place a week prior. So the district attorney puts Bob and Sam into separate rooms and offers them both the same deal.

  • If you both keep quite, you both get 1 year in prison for selling drugs.
  • If one of you rats out the other for the murder and the other stays quite, the confessor will go free and the other will get life imprisonment.
  • If you both rat each other out, you will both get 10 years in prison.

This scenario is called the prisoner’s dilemma and though it may seem like the obvious solution is for both Bob and Sam to stay quite, in reality both of them acting rationally will lead to a sub-optimal outcome. But we will get back to this.

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Numeracy #9: Regression Toward the Mean

In the late 19th century, Sir Francis Galton was studying how extreme characteristics (such as height) were passed on from parents to offspring. Galton observed that tall parents, on average, produced offspring that were moderately shorter, and short parents, on average, produced offspring that were moderately taller. Galton coined the term regression to describe the fact that parents who lie at the tail end of a distribution produce offspring that tend towards the middle of the distribution.

Today, this concept has wide-ranging applications from sports to finance, but human psychology hasn’t developed in a way to take advantage of it. Humans naturally think in terms of trends rather than statistical models. If something is working, we should do more of it, and going against that flow is counter intuitive. For example, the tech bubble of the late 1990s had a lot of very smart people believing that the mean itself had changed. This ended badly.

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Numeracy #8: Power Laws

In past posts, we explored how the normal distribution dominates our lives. Consider the heights of people in a given population: some will be extra tall, some extra short, and most will lie somewhere in between. Since no importance is given to outliers in these linear systems, we can compute some sort of average representation through sampling. The normal distribution guarantees us stable long-term results around that average. If the tallest person in the world walks into our sample, our average will only change by a few feet.

Now, consider what would happen if someone 50 billion feet tall walked into our sample. Or if someone worth $50 billion walked into a sample of people’s net worths. The average would change so much that the concept of it would become nonsensical. When values diverge rather than converge on an average, they are displaying the characteristic of a nonlinear system that follows a power law.

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