Math and probability are linked

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It probably won’t come as a surprise to most of my readers that math has always come fairly easily to me. What is sometimes a surprise is the type of math that I use mostly.

It generally falls into one of two categories. It is either algebra or combinatorial math. Once in a while I delve into some true statistics.

Some of you may be wondering where probability falls into all this as this is a term I use often. Probability is really a portion of combinatorial math. Combinatorial math is what tells us how many different ways five cards can be dealt from a 52-card deck. In turn, we can calculate how many Straights or Full Houses will be dealt in a similar fashion. Probability is just the number of ways a subset can dealt divided by the total ways. 

I have to admit that going all the way back to middle and high school I had a tougher time understanding how people didn’t see this type of math as simply as I do. Then I had a kid who was a middle schooler who is taking algebra. He does O.K., but clearly it doesn’t come to him as easily as it did me. Substituting X for Y seems obvious to me. Not as much to him. Working with him has given me some insight into the idea that not everyone gets this the way I get it.

I’d like to think that I do my very best to break it all down as much as possible in my column, books and even analysis reports for regulatory bodies. But every so often I’m given a reminder that the average person out there might struggle with concepts that are second nature to me.

This past week, I was reading a column about one of the COVID-19 vaccine trials. A family consisting of three people all signed up for the trial (for the Moderna version, I believe). As the trial is now over, they were notified that all three of them received the actual vaccine and not the placebo. As they were interviewing one member of the family, he remarked how he wondered what the odds were that all three of them got the actual vaccine. 

In the case of these drug trials, I believe that half got the vaccine and half got the placebo. So, the probability that all three had the same outcome is 25%. The probability that they all got the vaccine is half of this or 12.5%. There was an equal probability that they all got the placebo. The calculation is fairly simple — ½ x ½ x ½ = 1/8.

I’m sure most of you can see the parallel to a simple coin flip. What is the probability that you flip 3 heads in a row or 3 tails in a row? 12.5%. What is the probability that Red comes up on a roulette wheel 3 times in a row? This is only marginally trickier. There are 18 red numbers and 38 total numbers (assuming a double 0 wheel). So, the calculation is 18/38 x 18/38 x 18/38 = 10.63%. Hopefully, it is no surprise that the probability of 3 reds in a row is a little less than the probability of 3 heads in a row because the probability of one of them is a little less. 

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What I guess I found amusing about the story was that the person seemed to think that the probability would be far lower. But there really was only 1 other outcome — that two of them got one and the third got the other. This would make up the other 75%. 12.5% is simply not all that rare.

The probability of being dealt a Two Pair or better from a 5-card deal is 7.63%. We see this all the time when playing video poker and this is a bit more than half as likely as all three patients getting the same outcome in the trial.

About the Author

Elliot Frome

Elliot Frome’s roots run deep into gaming theory and analysis. His father, Lenny, was a pioneer in developing video poker strategy in the 1980s and is credited with raising its popularity to dizzying heights. Elliot is a second generation gaming author and analyst with nearly 20 years of programming experience.

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