It’s Not Too Early


I had the opportunity to re-connect with three former GMAT students from a couple years ago, and I found out something surprising: none of them have applied to business school yet.

This isn’t the situation where they took the GMAT on a whim without any intention of pursuing an MBA. All three still plan on attending business school. This also isn’t the situation where they’re worried that their scores aren’t good enough to get in. All three of these students scored in the 700s. This is simply students who planned ahead.

While they were still in school-mode, or close enough to it, they decided that within the next five years they saw themselves pursuing an MBA so they took care of the GMAT part of the equation in the gap between school and work or early enough in their careers that they had adequate time to devote to studying. Now they’re working on building their professional resumes so that when they come out of business school, they’ll not only have the MBA, but also the professional experience that will help land them coveted managerial jobs.

Scores from both the GMAT and the GRE are good for five years. That means if you take the GRE in 2014, knowing that you plan on doing some work and some travel until 2018, your scores from this year are still valid.

Of course, you accept some uncertainty by taking your test now. Maybe four years down the line, you won’t want to pursue graduate work. It’s certainly a possibility, however I think it’s outweighed by several other factors. First, your life tends to get busier with each passing year. I can’t think of how many times I’ve thought that things would calm down in the next year, but there always seems to be a flurry of new things popping up that eat up my time. Second, the closer you are to school, the easier it is to go into study mode. After a few years of working you get in a rhythm that says when you get home from work it’s time to relax. Trying to switch out of that into a mode that says you now need to sit down and work on your prep is tough. Third, you never know when an opportunity will pop up. If a good life or work situation suddenly goes bad, or you see a great grad school opportunity, you can grab it without having to wait for the several months the study and test process takes.

I’m excited for the work that my former students are going to do in business school and beyond, even though they aren’t there… yet.

 Is Warren Buffett’s $1 Billion Safe?


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!


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


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!