**Why Your Iowa Caucus Math is (Probably) Wrong**

**by Devon Cantwell**

*February 2, 2016*

The Iowa Caucuses are now the most recent example of why we need to revamp our emphasis on some statistics education in the US.

One of the most popular articles floating around today, in the aftermath of a brutal and tight Monday night at the Caucuses, is poor math analysis about how improbable it was for Clinton to win 6/6 coin tosses last night.

The most popular theory claims that the probability for this event is the following:

P(HHHHHH)= (½*½*½*½*½*½)= 1/64 or 1.56% probability of this event occurring.

This is not accurate.

To get a true understanding of the math behind whether this event was probable or not, we need to understand two major statistical tools: the Law of Large Numbers (LLN) and simulation modeling.

**Law of Large Numbers**

This central theory, one that I taught my AP Statistics students many moons ago, asserts that as you do more trials of an independent event (like a coin toss), your results should begin to norm and approach a limit– that of the expected value of the event. In the case of a coin, we would expect numbers to even out, starting at 30 samples, at about .50 for the probability of flipping heads in 30 or more samples. 30 is the minimum value for LLN to apply and for a sample or study to be considered “normal.”

In a study from 2004 by Bruce E. Trumbo in the Department of Statistics at California State University, Hayward, we are given a simulation of ten coin tosses. Here, *i* represents each coin toss occurrence. Toss represents the result of the toss, *i*. R*i *represents the actual value of our desired event (in this case, flipping Heads on a coin)

SOURCE: http://www.sci.csueastbay.edu/~btrumbo/Stat3401/Hand3401/LLNCoinTossB.pdf

As you can see here, the first five flips in this simulation result in H (Heads), with the last flip landing on tails. Through the simulation of ten coin tosses, we begin to approach our expected value of .5, but still land at .70, or in other terms, the coin landing on Heads 7 out of 10 of the times.

**Simulations and Sample Size**

The next piece to consider is: how common is it that the event of P(HHHHHH) will occur?

A November 2005 publication from Cornell helps us understand this below:

SOURCE: http://www.cs.cornell.edu/~ginsparg/physics/info295/mh.pdf

In the case of the simulations and example above, if we were to have flipped coins across 50 caucus sites 100 times in a row (just to be fair, you know), we would actually expect at least 27 of those sites to report 6 consecutive Head flips for their sample of coin flips.

If we had kept flipping this coin, it would have been far less likely that it would have continued to be Heads– that probability substantially drops after 8 consecutive heads, going from 17% to 8%. The probability begins to decrease almost exponentially (becoming about half each time).

**Conclusion**

The clear conclusions we can build from these mathematical understandings is that in the case of the Iowa Caucuses last night, our sample size was too small to rely on the expected value of .5 for the six coin tosses, and that all in all, the probability of this event across a large number of samples in this event would happen about 55% of the time. Perhaps the Sanders camp should switch to calling Heads for the rest of the the campaign trail.