When Is a 50:50 Coin Flip Not 50:50?

[What Would Jesus Do?]

What are the odds of getting a head when you flip a fair coin?

50:50. We all know that.

If you just got a head on a flip, what are the odds that your next flip will be a head?

That's 50:50 too, right?

Maybe not. Might be 40:60.

And what does that have to do with basketball shooting streaks?

http://www.wsj.com/articles/the-hot-hand-debate-gets-flipped-on-its-head-1443465711

 

[Run&Blade]

Everything is 50/50 either it happens or it doesn't.

 

[What Would Jesus Do?]

Everything is 50/50 either it happens or it doesn't.

I have a 50:50 chance of sleeping with Kate Beckinsale tonight? And another tomorrow night? And again the following night? Yee-haw - the odds favor me sleeping with Kate eventually.

 

[22*43*51]

Actually, flipping a coin is not a 50/50 proposition.

Generally speaking a coin has a bias(although slight) for landing in the orientation it was before it was flipped.

The type of coin matters too. A penny has a much greater probability to land tails-side up due to the amount of weight in Lincolns head.

I know. CSB

http://www.smithsonianmag.com/scien...-coin-flip-arent-quite-5050-145465423/?no-ist

 

[TexMichFan]

Thanks this will give me something to think about during my bike ride today.

 

[fredjr82]

Heads I win, Tails you lose. That's how

 

[Hoosierhawkeye]

Taking a bunch of unguarded shots is a very poor way to try to disprove hot hand when it comes to actual game time situations.

As they said there are too many variables in a game to be able to test that. What I would argue is that someone with a "hot hand" for some reason or another is better able to manipulate those variables in their favor.

 

[Huey Grey]

I'm guessing that this will be discredited. The comment section in the article already discovered several problems with the idea that a coin flip isn't 50/50. One simple problem could be related to limiting the coin flips to 4 flips. If the researchers would have say, counted up the number of heads in a series to 200 flips, it would have been much more compelling.

For example, I used a coin flip generator to flip 200 coins. 56 times a head followed a head and 44 times a tail followed a tail, bringing both numbers to 100 or exactly 50%. You can try it yourself and see what you come up with.

https://www.random.org/coins/?num=200&cur=60-usd.0001c

 

[Pepperman]

I've got to think about this for a while. Effectively, this academic paper is suggesting "independent" trials are not truly independent. But yet the greatest invention ever in a casino is the roulette tower that shows the previous results. People LOVE to believe the past influences the future, and this paper seems to curiously support that notion?

 

[Joes Place]

That's cool! It's a 'finite sequence' problem. That 40% number is only true for shorter sequences, but this is precisely what you are dealing with when considering how many 'shots' that are attempted by a guy in a game.

The link to the actual paper is here:
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2627354

 

[Huey Grey]

I ran three more simulations. The first yielded 45 heads following heads and 51 tails following tails. The second simulation yielded 45 heads and 38 tails. The third yielded 46 heads and 43 tails.

 

[Joes Place]

I ran three more simulations. The first yielded 45 heads following heads and 51 tails following tails. The second simulation yielded 45 heads and 38 tails. The third yielded 46 heads and 43 tails.

The paper is referring to the number of 'heads after heads' vs. 'tails after heads' in a finite sequence (<10 or <20). It's based on the number of possible permutations available, and it appears to be correct. That 40% number will eventually run up to 50% for an 'infinite' sequence, but for small sequences, it starts out at about 40% and moves up with the sequence length.

 

[Huey Grey]

And one more round to bring it to 1000 flips: 37 heads and 52 heads. Which brings the grand total to 229 heads following heads and 228 tails following tails. For 40% you would expect 200 heads and 200 tails and for 50% you would expect 250 heads and 250 tails.

 

[Joes Place]

This makes me want to try this out on a Roulette wheel in Vegas on the next trip. Although those odds are only 49% (because of green 0/00), I'm wondering if the effect might be similar for small runs of bets...

 

[Joes Place]

And one more round to bring it to 1000 flips: 37 heads and 52 heads. Which brings the grand total to 229 heads following heads and 228 tails following tails. For 40% you would expect 200 heads and 200 tails and for 50% you would expect 250 heads and 250 tails.

No. You need to run it for 'one million' (or just 'a lot of') sets of '4 flips in a row', not a continuous data set....the longer the total series, the closer you will converge to 50/50

 

[naturalmwa]

Actually, flipping a coin is not a 50/50 proposition.

Generally speaking a coin has a bias(although slight) for landing in the orientation it was before it was flipped.

The type of coin matters too. A penny has a much greater probability to land tails-side up due to the amount of weight in Lincolns head.

I know. CSB

http://www.smithsonianmag.com/scien...-coin-flip-arent-quite-5050-145465423/?no-ist

I always suspected the game was rigged.

 

[Joes Place]

I have a 50:50 chance of sleeping with Kate Beckinsale tonight? And another tomorrow night? And again the following night? Yee-haw - the odds favor me sleeping with Kate eventually.

But, your chances of sleeping with her the 2nd night drop to 40%,per this paper's results....

 

[TexMichFan]

I'm guessing that this will be discredited. The comment section in the article already discovered several problems with the idea that a coin flip isn't 50/50. One simple problem could be related to limiting the coin flips to 4 flips. If the researchers would have say, counted up the number of heads in a series to 200 flips, it would have been much more compelling.

For example, I used a coin flip generator to flip 200 coins. 56 times a head followed a head and 44 times a tail followed a tail, bringing both numbers to 100 or exactly 50%. You can try it yourself and see what you come up with.

https://www.random.org/coins/?num=200&cur=60-usd.0001c

I am just impressed you have a coin flip generator.

 

[Huey Grey]

No. You need to run it for 'one million' (or just 'a lot of') sets of '4 flips in a row', not a continuous data set....the longer the total series, the closer you will converge to 50/50

But if we just limit it to sets of 4, we're rigging the outcome. Why not sets of 10? Or sets of 20? Some nights, players like Kobe or Jordan put up close to 50 shots. If you're tying to base it on the number of shots put up in one night, 20-30 attempts would be a lot more likely for someone with a hot hand than just 4 attempts.

And here's where I have a problem with the article. You shouldn't even limit yourself to even 20-30. The shots should span across multiple games and even multiple seasons. A HOF player will put up some 20,000 shots over a career. With that many shots, you would expect long runs of at least 10 shots made in a row during multiple points of that player's career. It doesn't mean that they have a hot hand that night. It just means that the randomness of their expected makes beat the odds on that particular night.

 

[What Would Jesus Do?]

The paper is referring to the number of 'heads after heads' vs. 'tails after heads' in a finite sequence (<10 or <20). It's based on the number of possible permutations available, and it appears to be correct. That 40% number will eventually run up to 50% for an 'infinite' sequence, but for small sequences, it starts out at about 40% and moves up with the sequence length.

Seems to be merely an artifact of the clustering rules they imposed. Or dictated by those rules might be a better way to put it.

They are looking at clusters of 4 "tosses" and at least 1 of the 1st 3 has to be a head. They then look at the toss after the head.

On average, all clusters of 4 tosses will have 2 heads. So if you preselect clusters of 4 that already have at least 1 head and look only at the next toss, it makes sense that any other toss in that cluster has a higher probability of being a tail.

 

[Pepperman]

This makes me want to try this out on a Roulette wheel in Vegas on the next trip. Although those odds are only 49% (because of green 0/00), I'm wondering if the effect might be similar for small runs of bets...

No, it won't. Because the "next" trial is totally random.

What they did is look at coin flips "in sample".

 

[DanL53]

Screw the math. What interests me is a Hall of Fame player putting up 20,000 shots. Supposing he only makes about $100,000,000.00 over the course of his career? Which sounds low to me, but I don't follow Pro-Basketball. That's only $5,000 a shot! Or if he makes half his shots, $10,000 per make.

Now, I think it would be fun to do away with the current method of keeping score and come up with a way to track points per money spent to earn them. So, it would prove beneficial to have a bucket scored by a lower paid player! A player making minimum wage would then prove highly valuable! I'm sure people get my idea without further explanation.

We'll call this, the CBL. "Corporate Basketball League".

 

[Pepperman]

Dan, the NBA already is "corporate" and believe me when I say they have already been doing the math as you suggest. And the result of your hypothetical experiment is called the "NBA".

 

[What Would Jesus Do?]

But, your chances of sleeping with her the 2nd night drop to 40%,per this paper's results....

Even those odds have me excited.

 

[Joes Place]

But if we just limit it to sets of 4, we're rigging the outcome. Why not sets of 10? Or sets of 20? Some nights, players like Kobe or Jordan put up close to 50 shots. If you're tying to base it on the number of shots put up in one night, 20-30 attempts would be a lot more likely for someone with a hot hand than just 4 attempts.

And here's where I have a problem with the article. You shouldn't even limit yourself to even 20-30. The shots should span across multiple games and even multiple seasons. A HOF player will put up some 20,000 shots over a career. With that many shots, you would expect long runs of at least 10 shots made in a row during multiple points of that player's career. It doesn't mean that they have a hot hand that night. It just means that the randomness of their expected makes beat the odds on that particular night.

But that is the point. It's a small finite sample problem, for short sequences.

If you run your simulator and only use a row of 6 flips as each 'sample', only pick the rows which start in a 'heads'; then count how many H-H vs H-T there are in that 6 sample sequence. You will see that for ~ 10 trials (just keep going down the page and pick the rows starting with a H) that the H-T vs H-H is about a 60/40 on average ratio. That ratio will converge to 50/50 the longer the sequence you choose.
Don't use overlapping sequences.

This has nothing to do with large samples over a career, because that violates the assumptions in the paper - it is for short runs. Sure, a player can put up 30 shots in a game, but more typical is 10-20 shots max.

 

[tarheelbybirth]

But that is the point. It's a small finite sample problem, for short sequences.

If you run your simulator and only use a row of 6 flips as each 'sample', only pick the rows which start in a 'heads'; then count how many H-H vs H-T there are in that 6 sample sequence. You will see that for ~ 10 trials (just keep going down the page and pick the rows starting with a H) that the H-T vs H-H is about a 60/40 on average ratio. That ratio will converge to 50/50 the longer the sequence you choose.
Don't use overlapping sequences.

This has nothing to do with large samples over a career, because that violates the assumptions in the paper - it is for short runs. Sure, a player can put up 30 shots in a game, but more typical is 10-20 shots max.

I might be missing something but this looks like one of those math tricks where you add numbers to some number and multiply and divide and reverse the numbers and you end up with your age. There is nothing about the first flip of a coin that can have an impact on a subsequent flip to influence the outcome. A coin flip - assuming a fair coin - is going to be 50/50. It sounds like they've imposed conditions on what they count and those imposed conditions are creating the anomalous result. But I hated sadistics in college so I'm not about to delve into their paper.

As for the hot hand controversy, screw the statisticians. We've all seen a player get hot and hit 10 in a row from deep. I've never viewed it any differently than flipping a coin 10,000 times and getting 10 heads in a row. That the averages will always play out and there will be a poor shooting night doesn't change the fact that for THAT game the player had a hot hand.