Is Warren Buffett’s $1 Billion Safe?

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March is my favorite part of the year, and the NCAA March Madness is no small part of that. For those of you that are unfamiliar, after months of regular season basketball, weeks of conference tournaments, and two days of preliminary matches the field is narrowed to 64 teams. Those teams are put into a single-elimination bracket with four regions, and after 63 games a champion is crowned.

This year’s tournament has a little extra excitement as businessman Warren Buffett has offered $1,000,000,000 to anyone who can complete a perfect bracket. But Buffett didn’t luck into his fortune. Let’s use basic probability to determine just how likely it is that someone will be collecting that billion dollar check!

First of all, we need to determine whether we’re working with independent or dependent events. In this case, our outcomes are independent. The result of one game does not alter the odds in the next game. Dayton beating Ohio State doesn’t change the likelihood of Harvard beating Cincinnati. In contrast, a dependent event would be something like pulling marbles out of a bag without replacement. If we pull a blue marble out of the bag there is one less blue marble to pull out the next time, so the odds will change.

With independent events, we can compute the probability of all n events in a series happening by the following formula:

P(1)* P(2)* P(3)*…*P(n) = Probability of all events occurring

In this case, we need to know the probability of predicting all 63 events correctly. In our first case, let’s assume we know nothing about college basketball. In that case, we’ll randomly guess a winner from each matchup. Since we know nothing about the teams, we will estimate our likelihood of getting each guess correct at 1/2 or 0.5. In that case our likelihood of predicting a perfect bracket will be 1 in 2^63 or 1 in 9,223,372,036,854,775,808. That’s more than 9 billion billions! So without any knowledge it’s safe to say that your chances are not good.

But what if you know a little bit more? What if you know that a 1 seed has never lost to a 16 seed? That makes four games slam dunks and just leaves 59 left to figure out. Maybe you’re a college basketball expert and you can predict 60% of outcomes correctly. Your chances of getting a perfect bracket are now 1/(0.6)^59 or 1 in 12,275,963,663,147. We’re down to one in twelve million billion, but Mr. Buffett’s money is still looking pretty safe.

Let’s try one other example. Let’s say you’re a college basketball expert with a time machine that allows you to look two days into the future. That will allow you to know the first 32 results with absolute certainty, and just leave the final 31 games that you can predict correctly 60% of the time. Now your chances are 1/(0.6)^31 or 1 in 7,538,956. Even with expertise and a time machine your chances are just one in seven and a half million!

Three quick takeaways:

1. When events are independent, multiply the probabilities to find the total probability of all events happening

2. Exponential growth makes numbers get really big (or really small) really fast

3. Warren Buffett’s money is probably safe… until we build a better time machine!

 

 Happy Pi Day!

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In honor of Pi Day, here are 3.14 things you need to know!

1. Pi Day is celebrated in the United States almost exclusively by math nerds (although the occasional desert lover will jump on board as an excuse to partake of some banana creme or lemon meringue). Math nerds in other countries don’t get the same opportunity to celebrate though! That’s because the American date format (month/day/year) gives us 3/14 every year on March 14th. However, because there is no 14th month nor an April 31st, the international date format (day/month/year) doesn’t allow a 3/14 (or 31/4)!

2. Pi is defined at the ratio of the circumference of a circle to its diameter. It’s not a concept that was created by mathematicians, but rather something that was discovered as a fact of nature. No matter the size of the circle this ratio between circumference and diameter will stay constant! It’s a magic number that’s found in nature and that’s part of what makes it so attractive to math fans.

3. Pi is an irrational number. That doesn’t mean it makes bad decisions or gets upset at trivial things! An irrational number is one that can’t be defined as either a decimal or a fraction. Though you’ll often hear pi referred to as 3.14 or 22/7, neither one of those is exactly right. Pi’s first 10 digits are 3.1415926535, but that’s just the beginning! The digits of pi will go on and on and on infinitely without falling into any repeating pattern!

3.14. While pi is an irrational number that can’t be completely accurately stated in decimals, if you need to approximate on your test 3.14 will always be accurate enough for what you need, so don’t worry about memorizing pi out to the 100th digit… unless of course you want to!

Happy Pi Day!

Three Approaches to Average Sleep

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As a new parent, sleep is one of the most important commodities in my life. Sympathetic friends often ask how much sleep I’m getting. If I want to respond with the amount of sleep I’m getting “on average” there are several different approaches I can take. Consider the following data:

The D row tracks the days being studied. The H row tracks the longest stretch on uninterrupted sleep on that day (in hours).

If we take the traditional meaning of average and take the mean, we add all the totals together and divide by the number of days. When we do that we find that I had an average of 4.44 hours of sleep per night. But that doesn’t tell the only possible story. Perhaps you can see from the data that my wife and I alternate nights on baby duty. On odd numbered nights I tended to get less sleep and on even numbered nights I tended to get more sleep. Although 4.44 is the mean, there is quite a bit of spread around that number.

A second approach to average would be to take a median. Perhaps that would give a better picture. When we take a median we simply take the middle term of the set, or when there is an even number of terms the average of the two middle terms. For this full set, that yields a median of 4.75 hours- a slightly rosier picture of my sleep situation. However, in a set like this the selection can influence the results. If we take the median of days 1-7 we get 4.5 hours and if we take the median of days 2-8 we get 5 hours. Trust me, those 30 minutes can make a lot of difference.

A third approach to finding average can be to find a mode. A mode is simply the most common number found in a set. In this case, I got 5 hours of sleep twice over this 8-day period making the mode 5. However, if we ignore all half hours and simply round everything down, we would have 2 modes: 2 and 5.

Here you see three different ways that we can discuss a data set in order to find an average or a middle. Hopefully this discussion has helped you to see how they work, how they can give different results, and what those results mean!

 Post-Holiday Discount Candy

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I have a sweet tooth. I have a budget. These two facts are often at odds with each other when I’m out shopping. I was out at the drugstore the other day when I noticed the discounted Valentine’s Day candy in the heart-shaped boxes.

 

Now, perhaps you think it’s silly for a man to by himself candy in a heart-shaped box nearly two weeks after Valentine’s Day, but I am nothing if not a pragmatist. If I was getting the right value, I was ready to buy.

 

Value in the pre-holiday market is almost impossible to find. That’s because of the large mark-up on the candy due to decorative wrapping. But, as a non-sentimental pragmatist I get no value out of the wrapping. Let’s say that there is a 50% mark-up on all candy in decorative wrapping. (The actual number is probably higher, but for the sake of this example and for the sake of not sending me into a rant about inefficiency and marketing, let’s say it’s 50%). For instance, if the fair price of the candy is $2, it will be priced at $3. The question is this: if the fair value of the candy is marked up by 50%, how much does the candy need to be discounted in order for me to get a fair price?

 

Many of you probably defaulted to a quick answer of 50%. Let’s take a moment and see why that’s not correct.

 

Let’s call the fair value of the candy f. If we increase that by 50%, we get the original price at the store: 1.5f. Now, if the store decides to put candy on sale for 50% off of the sales price, you end up paying 1.5f * 0.5 = 0.75f. So, a sale of 50% actually allows you to pay a price below the fair price… a good deal! If we want to figure out the discount we need from the store in order to get a fair price, we use the following equation:

 

1.5f * (1-y) = f

1.5f – 1.5fy = f

1.5fy = 0.5f

1.5y = 0.5

y = 1/3

 

So, if the candy is 1/3 off, or approximately 33% off, you’re getting a fair price. This answer that may not have been intuitive comes from the fact that in a percent change problem you always need to identify your base price. The first time we increase the price by 50%, we’re using the fair price as our base. When we decrease the price by 50%, we’re using the store price as our base. That’s a big difference and something to keep in mind the next time you face a percent change problem… or go to buy some post-holiday candy!