#49) Flipping Pennies
[Note about this transcript -- it is approximately the same as what is said in the video -- the same material is covered, though the video might be more fun to watch.]
[Mel starts at the white board, writing notes as he goes along.]
From 1980 to 2016 (10 elections), there were 19 counties that successfully voted for the candidate who won the presidency. That is, they all voted for Reagan in the 80s, then Bush Senior, Clinton, W Bush, Obama, and finally Trump in 2016. An amazing feat. These are called bellwether counties, because they seem to "predict" the national outcome. Perhaps they are purple and more sensitive to the overall national trends -- nobody quite knows what makes these counties so special. They're just really good at picking the winners.
But then something strange happened in 2020. 18 of those 19 bellwether counties voted for Trump. Yeah. Only 1 county got it right (11 times in a row). The argument is that this is such a statistically impossible result that fraud must have occurred. I'll show you the math in a bit, but first, it's time to perform our own experiment, and I strongly recommend watching this till the end. I promise that it'll be worth it.
[Cut to the kitchen table -- family members wait around the table while I explain.]
Here I am with my family at the kitchen table and we're going to simulate bellwether counties. But instead of counties, we're going to be flipping pennies. I have here 1000 freshly minted pennies -- never opened and ready to test.
As you may know, when we flip coins, it's about 50% heads and 50% tails, but that's only if the coin is perfect. Small imperfections can cause a coin to be biased, and that's what we're going to test for today. We are in search of biased coins! And since 1000 is considered to be a hefty statistical sample, we're surely going to find at least one biased coin in this batch.
Okay, I'm going to divide out the coins, and we'll start flipping. Along the way, I'll collect the coins in these buckets. Heads means "win," and I'll collect those in this blue bucket. And tails means "lose." I'll collect those in this orange bucket. And then we'll count the results. How many do you think will come up heads? Let's find out.
[Time lapse video of the first round of flipping, collecting, and counting.]
Okay -- we're done with the first round. Looks like we ended up with 494 heads. That's pretty close to 500. How many of you guessed that number? Well, we're going to do another round of flipping. How many of these coins will end up heads again?
[Time lapse of round 2 of flipping.]
Okay -- back again. We now have 271 coins that landed on heads twice in a row. Note that these are all 2022 coins -- in the name of science they are basically the same, but we'll find the biased coin. We're going to keep flipping these coins.
[Time lapse of 5 more rounds of flipping.]
You're not going to believe this, we've done five more rounds, and we still have some pennies that are coming up heads every time. After round 3 we had 144. After round 4, it was 76. Then 26, and 16, and finally 9. That's 7 times in a row, and we're down to just these few pennies here. I guess we'll keep going. We'll do the rest in real time here.
[Video continues with us flipping in real time until it comes down to the last penny in the 10th round. Note -- we make clear that any coins that fall off the table must be re-flipped.]
Wow! Can you believe it? This penny that I'm holding right now has flipped heads 10 times in a row. That's just crazy. I'm getting nerdy math chills right now just thinking about it. It looks like we've found our biased penny. Come over here to the whiteboard, and I'll show you the math so you can see exactly how exciting this is.
[Mel goes back to the white board.]
Probability that 1 coin flips head 10 times in a row = (1/2)^10 = 0.000976. On a normal curve, this shows up at about negative 3-sigma. Anything under 5% likelihood is a statistically significant event. This proves that this one coin is indeed a biased coin.
What about the 18 out of the 19 bellwether counties that failed to predict the presidential winner? Assuming each county has a 50% chance of guessing correctly, the probability of all 18 counties all being wrong at the same time is (1/2)^18 = 0.0000038: an incredibly significant unlikely anomaly. There's definitely something going on here.
[Mel goes back to the kitchen table]
Wanna see how many times we can keep flipping this coin heads?
[Mel and family flip the biased coin 10 times, but it keeps coming up tails. We see this: TTHTTHHTHH.]
I don't understand. This coin did heads 10 times in a row, and then we flipped it 10 more times and it went back to 50%. I must have done something wrong with my math.
[Time lapse of Mel rechecking his math, crossing stuffs out and figuring out the error.]
Um, you're not going to believe this. I've made a huge mistake in my math -- but I fixed it. Let me show you.
[Mel goes to whiteboard.]
Okay -- how do I explain this? It's very rare for a coin to flip heads 10 times in a row. The chance of flipping one coin heads 10 times in a row really is 0.000976. But we didn't flip 1 coin 10 times in a row. We flipped 1000 coins. Remember how 494 flipped heads the first time, and then the number dropped by half with each subsequent round? Well -- we kind of already expected this to happen. Because we flipped so many coins repeatedly, we basically asked for the impossible to happen. 1/2 raised to the 10th power rounds to 0.001 -- meaning it would happen 1 out of 1000 times. And since we flipped 1000 coins, that means we would expect about 1 coin to flip heads 10 times. See where I'm getting at?
Number of Coins that Flipped Heads:
| 1: | 494 | 6: | 16 |
| 2: | 271 | 7: | 9 |
| 3: | 144 | 8: | 3 |
| 4: | 76 | 9: | 2 |
| 5: | 26 | 10: | 1 |
Now watch this -- this is crazy. What are the chances of at least one coin flipping 10 times out of 1000 attempts? We can calculate the opposite of that pretty quickly. That is -- what are the chances that all 1000 coins don't flip heads all 10 times? For one coin, the chances of getting at least 1 tail is the opposite of this number here [0.000976], which is [0.999024]. And since we have 1000 independent coins, we raise this to the 1000th power -- and wow -- that brings it down to 0.376. That is 37.6% chance of all coins getting at least one tails each. And so, the opposite is ... 62.4% chance of having at least 1 coin out of 1000 flip heads 10 times in a row. That's a crazy high probability of such a rare event occurring!!
At this stage -- you're probably figuring out that earlier I messed up the math on purpose. This is because I wanted to capture the essence of the mistakes people make when they formulate similar bellwether arguments. It's an example of using math that looks correct, but being applied incorrectly. This mistake is an example of survivorship bias -- where we errantly endue magical prediction properties on something solely on the fact that they had the dumb luck of being correct so many times in a row.
Not convinced? Well -- check this out. We also did a losers bracket [time-lapse of more flipping in the background]. That is, we took all the tails from the first round, and flipped them all again, only this time keeping all the tails. And we ended up with a coin that flipped tails 9 times in a row. Here are the numbers we saw from each round. Notice how the numbers likewise follow a half-by-half pattern, just like for the heads run. We also flipped that last coin 10 times, and it likewise came up tails only 5 times. 50%!!
Number of Coins that Flipped Tails:
| 1: | 510 | 6: | 21 |
| 2: | 237 | 7: | 10 |
| 3: | 126 | 8: | 3 |
| 4: | 66 | 9: | 1 |
| 5: | 38 |
Still not convinced? Feel free to test it out, yourselves. Spend $10 for 1000 pennies. It would be a fun family night. See if you get similar results.
And still still not convinced? Okay -- here's another analogy -- the same phenomenon. Let's say we're at a baseball game, and the seats are jammed pack. Oh look -- here I am -- this red dot right here. What are the chances that I'm going to catch a fly ball? Not much of a chance -- right? Those balls could fly anywhere in the park.
So, now we see someone hit a foul ball. It's headed toward this section right now. Am I going to catch it? Probably not. If there are 1000 people in this section, I only have a 0.1% chance of getting lucky. But look at the ball. We can see that it's going to land somewhere in that section. There's a 100% chance that someone is going to catch this ball. So, how does 0.1% chance become 100%? Only because there is a large number of data points. And is that one person special? Not really. Just lucky.
Likewise -- with 3,143 counties, we have a whole bunch of datapoints -- so we'd expect to see "bellwether" counties in the results.
What about that one county that's still a bellwether? Let's take a closer look. It is Clallam County in Washington (the most western county in the 48 contiguous states). If we look back in time, we see that they voted incorrectly in 1976 and 1968, 1916, 1912, 1896, and 1892. That's 6 out of 33 elections wrong. An 81.8% success rate? I guess that isn't too shabby. But still not fully reliable.
Let's do the math again and see what the chances of an 81.8% success rate is for one county out of 3,143.
Calculating this is beyond the scope of this video, so you can ask a math friend if I'm doing this right. We'll use the null hypothesis that no county is special, and try to disprove it. So, we assume the chances of being correct is 50%. To calculate the chances of getting 27 or more elections correct out of 33 contests, here's the formula, we need to add up all the chances of getting 0 wrong, 1 wrong, 2 wrong, all the way down to 6 wrong.
We must add these numbers:
0 wrong = 0.00000
1 = 0.00000
2 = 0.00000
3 = 0.00000
4 = 0.00000
5 = 0.00003
6 = 0.00013
Total = 0.00016
Like we did before, we'll take the opposite: the chances of being wrong more than 6 times out of 33 is 0.99984. There are 3,143 counties. So we raise this to the 3,143rd power to get the probability that ALL of the counties are wrong more than 6 times. This comes out to 0.60. Taking the opposite of that, the chances of at least one county getting 6 or less wrong is about 40%. That's crazy! It's impressive to get 27 elections correct, but because of the large number of counties, the chances of this happening due to dumb luck is within the realm of possibility.
This is the beauty of survivorship bias. When we look at just one county, we see an incredibly low chance of 0.016%, but when we look at all 3143 at the same time, that probability raises to 40%. This is the baseball landing in the hand of a non-special person in a large crowd. If you have a large number of data points, the impossible naturally becomes possible.
And there is nothing special about Clallam County. It's just dumb luck.
I hope you've enjoyed this special mathy experiment. There's a lot more where this came from. And I'll see you here next time on "Always Be Better."
Addenda: Here is some information about the math -- not included in the video.
Assumption #1) The 3,143 U.S. counties have all existed for all of the last 33 elections. This isn't quite true, but a more precise calculation wouldn't change the answer much.
Assumption #2) Each county has a 50% chance of voting for the president that wins. This also isn't quite true. Most counties are rural, so most are going to vote Republican in most elections. In this way, Democrat presidents tend to break bellwether counties, while Republican presidents tend to help them. On average, Republican vs. Democrat presidents are evenly spaced, so it's somewhat reasonable to expect that this effect would cancel out over time. Though I wouldn't be surprised if a more complex calculation provided a final probability as high as 80% or as low as 10% -- I think we'd still find the answer to be in the realm of possibility.
I don't have time to perform the full calculation, but you're welcome to try it -- I'd be interested to see the results, and critique the method.
Also, I've read, but haven't confirmed for myself, that if we count bellwether counties from 1976, we see a similar pattern where in 2016 we lose approximately 18 bellwether counties leaving only 1 correct. I would believe this result as 1976 was a Democrat year. Yet, how many cried "fraud" in 2016?
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