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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