Happy Holidays! As many of my readers might be in a light hearted mood this week, I decided to keep this column that way as well. I won’t discuss Straights or Flushes or even partial Royal Flushes. I’m going to discuss birthdays!
The topic is courtesy of my young son. You see, while driving him to school this week, he remarked how the other two classes in the same grade he is in were having birthday celebrations on the same day and that seemed “odd” to him.
But, when we explained to him it is possible they were celebrating someone’s birthday that was happening over the coming winter break, it didn’t seem so unlikely anymore. My son appeared to do some math in his head and then pronounced that he thought the odds were about 1/3 a kid in each class had a birthday during this period. As he is still in elementary school and hasn’t really learned much in the way of probabilities, I cut him some slack.
But that didn’t stop me from giving him a math lesson on the ride to school. The goal was to approximate the probability one kid in a class of 20 has a birthday over winter break. This is where we all got a lesson. Make sure you’re all talking about the same thing. In my mind, I was thinking about the kids who had their party in school that day and my brain counted a three week period – the final week of school plus two weeks of break. My son took into account only the two weeks of break. For simplicity, we went with his definition and used 14 days.
What is my quick shortcut for approximating the probability? We have 20 kids and 14 days. So multiply one by the other and get a total of 280 out of 365 possible days. Using some of his approximating skills he learned in school, he came up with about 75%. Not bad. This method will exclude cases where two-plus kids have a birthday in the two-week period. But it’s not a bad answer for sitting in the car without a calculator, right?
What is the exact answer? With the help of a spreadsheet or calculator it becomes rather easy. There are 365 days in a year (forget Leap Day for now). We’re looking at 14 of them. So, we were in the negative. What is the probability all the kids have their birthdays not in those 14 days? There are 351 days not part of our 14 days. So multiply 351/356 by itself 20 times (for each kid) and we get 45%. But, this is the probability of no one having a birthday during that period. To get the probability of at least one child, we subtract our result from 1 and get 55%.
That’s a pretty big spread between 55% and 75%. Is my shortcut method that bad? Something doesn’t seem right. Being me, I decided to investigate a bit. I started by using my method for only two children. So, now we have 28/365 or 7.7%. Calculating it exactly, I get 7.5%. That’s not too bad. I continued increasing the number of children. With three children my shortcut got 11.5% and the actual was 11.1%. With five5 children, my shortcut gives a result of 19.2% and the real value is 17.8%. Well, this is still close, but I’m beginning to see a pattern.
I move to 10 children and my shortcut gives me 38.4% probability. The actual calculation gives me 32.4%. I think you can see where this is going. By the time I get to 20 children, my difference is about 20% and all of a sudden my estimate is less of a quick shortcut and more of a wild guess. How is this relevant to us? It’s not. Just kidding! It teaches us to be careful about relying too much on short cuts.
For example, what if we use the same methodology for the probability of a Three of a Kind turning into a Four of a Kind in video poker. There is one card that can do it, we draw two cards out of 47, so we ballpark at 2 out of 47. This gives us an exact match. But, if we use this same method to figure out the probability of going 5, 10 or 20 hands without hitting the quads, we fall into the same trap. At 10 hands, using this method, we would say 20 out of 47 chance of hitting Quads, which is 42.6%. Calculating it mathematically, we get only 35.3%. At 20 hands, we get 85% with my shortcut and only 58% mathematically.
This becomes important if it starts affecting us. If we begin to believe what his happening is insanely rare, then we might no longer believe any of the math and the strategy it dictates. In reality, the problem is not the math, but our misunderstanding of the math. After all, what probability would I have gotten if there were 30 kids in that class (30 x 14 = 420. Divide this by 365 and get over 100%) Clearly, this would be proof positive the shortcut begins to breakdown.