The coin flip conundrum - Po-Shen Loh

When you flip a coin to make a decision, there's an equal chance of getting heads and tails. What if you flipped two coins repeatedly, so that one option would win as soon as two heads showed up in a row on that coin, and one option would win as soon as heads was immediately followed by tails on the other? Would each option still have an equal chance? Po-Shen Loh describes the counterintuitive math behind this question.

Lesson by Po-Shen Loh, animation by Augenblick Studios.

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[psloh]

Hi everyone! Thanks for your comments about the video. This is a really interesting paradox because there is a critical nuance: there are two separate coins being flipped.

If there was only a single coin being flipped, then as many of the comments indicate below, the game would be fair. However, if each person flips a coin separately, then the game suddenly becomes unfair. The mathematical analysis in the video helps to dig into what is really going on, and the game board displayed in the middle of the video makes the intuition more clear. The many comments on this topic show that this paradox is really counterintuitive, because the apparently small change from one coin to two makes a critical difference. If you still are doubtful, try writing a computer simulation which checks what is the average number of coin flips before you get Heads-Heads immediately in a row, and what is the average number for Heads-Tails. The experiment will match the calculations!

Ultimately, that is the value of math: reality is complicated, and often defies human intuition. Math gives us a framework for clearly analyzing what is correct! Thank you all for watching and commenting.

[andrewhoward]

I was very confused until I went back to the start of the video and realized they were flipping different coins.

[reegodlevska]

You oversimplify so bad that it gets complicated.

[thebigbadned]

You somehow managed to complicate the explanation by trying to oversimplify it.

[gaganahuja]

its quite simple...*easy explanation*: we firstly have to toss coins to the time it shows head...then
case1...if there is head we won but if not then we need to start again to get consecutive heads..
but in
case2 .. if we get tails we will win nd if not then it must be a head so we need not to start again..nd just have to get tails for once...

[shawnmccod]

Number-crunching - Here are some *cumulative* probabilities for winning within a given number of flips:
*Orville (Heads-Heads)*
*If his first flip was H* -
2 flips: 50.00% | Ways to win in exactly 2 flips (starting with H): {HH}
3 flips: 50.00% | No way to win in exactly 3 flips (starting with H)
4 flips: 62.50% | Ways to win in exactly 4 flips (starting with H): {HTHH}
5 flips: 68.75% | Ways to win in exactly 5 flips (starting with H): {HTTHH}
... (remember these probabilities are cumulative)
*If his first flip was T* -
2 flips: 0%
3 flips: 25.0% | Ways to win in exactly 3 flips (starting with T): {THH}
4 flips: 37.5% | Ways to win in exactly 4 flips (starting with T): {TTHH}
5 flips: 50.0% | Ways to win in exactly 5 flips (starting with T): {TTTHH, THTHH}
...
*Wilbur (Heads-Tails)*
*If his first flip was H* -
2 flips: 50.00% | Ways to win in exactly 2 flips (starting with H): {HT}
3 flips: 75.00% | Ways to win in exactly 3 flips (starting with H): {HHT}
4 flips: 87.50% | Ways to win in exactly 4 flips (starting with H): {HHHT}
5 flips: 93.75% | Ways to win in exactly 5 flips (starting with H): {HHHHT}
...
*If his first flip was T* -
2 flips: 0%
3 flips: 25.00% | Ways to win in exactly 3 flips (starting with T): {THT}
4 flips: 50.00% | Ways to win in exactly 4 flips (starting with T): {THHT, TTHT}
5 flips: 68.75% | Ways to win in exactly 5 flips (starting with T): {THHHT, TTHHT, TTTHT}
...
They both have an equal chance of winning in exactly two flips, but Wilbur has a greater chance of winning in exactly 3 flips, 4 flips, etc., so on average (and especially if both of them were unlucky to begin with), Wilbur will win in fewer flips.

[sebastianelytron]

I _still_ don't get it, I think I'm gonna *flip* out!

[NXeta]

1:05 holy damn that head zoom scared me

[alansands]

originally I thought each brother would flip 2 coins, (or the same coin twice and keep track of results), and they either got the desired result or they didn't. Then they would start fresh with another 2 flips. In that case, the probabilities would be equal.
But each brother flipping UNTIL they got their desired result, really does favor HT.

[danielp]

Every single TED-Ed video is like a perfectly scripted essay, hitting all the main points with supporting details and engaging writing style. It is amazing that so many people put so much work into these videos just to educate us about topics we might never have otherwise even considered.

[MrSinner]

Im watching this instead of studying for my trig exam.

Edit (2 hours later): Still havent bothered. Heading to class, exam in 40 (good thing im just auditing this course lol)

Edit: I answered 4 of 15 problems lol

Edit: it’s been 2 years or so but I’ve only moved 2 semesters forward in my maths. Calc 3 next term

[Resare]

I think the conundrum is somewhat misleading. If the question was "who has the most points after a large number of flips" with each brother earning a point when ever their corresponding "HH" or "HT" combinations occurr, than yes the brother with "HT" would have the statistical advantage and earn more points, but as it stands in order for the second brother with "HH" combination to go back to start, and thus loose his progress towards the win, a "HT" combination would have to occur which would stop the game immediately as the "HT" brother would have won.

[TheBetterVersion]

The interesting thing is that if they throw only one coin, then the 2 brothers would have equal chances. That's because we wait for the first head, and the coin after it determines who wins

[AA-]

Clarifications to everyone who suggests that both players should have an equal chance of being the first to flip a sequence of HH compared to HT.

Each player is flipping their own coin and the winner is the person who achieves their goal (HH or HT) in the fewest number of flips of their *own* coin.
The point is that if your goal is to achieve a sequence of HH with your own coin for example, you DO NOT automatically lose if you happen to flip a sequence of HT (Opponent's goal). The same rule applies that you don't win automatically if your opponent flips a sequence of HH with *his* own coins.
The common misconception (Probably because the rules weren't explained clearly enough in the video) is that the game is only played with 1 coin which applies to both players. In that case both players are equally likely to win as after a Heads is flipped, there is a 50% of the next coin being flipped as Heads or Tails, and thus achieving the respective player's goal

[garfunky]

Ok guys for those of you who were initially confused like me, there is a crucial bit of info in the video that wasn't made clear: Both brothers are flipping their own coins until one person gets their desired sequence. This whole time I thought they were just flipping ONE coin. The script doesn't make it clear but the visuals do. Hope this helps

[andyTONYpandy]

Can't believe how many people didn't pick up on the two coins but I think there is an interesting scenario where you can play the game with ONE coin and it will trick you.

I know it's a bit late to add this to the discussion but I think it works and wait to stand corrected.

Imagine 4 players (still works with 2 but 4 is more interesting) flip the same coin until one of the players hits their own chosen 3-coin combo.

Player A: HHH, Player B: HTH, Player C: TTT, player D: THH

On the face of it, it seems fair; each combo has a 1 in 8 chance. If the first flip is H, A and B are off the marks, if it's T, C and D. Then the second flip decides who advances and one more flip gives them a 50/50 chance of finishing. Note there is no possibility of a coin flip with 100% chance of a winner as there would be with TTH and TTT.

In reality, players A and C have a huge disadvantage. Every time they hit the wrong coin they go back to the start and have a 1 in 8 chance of getting the combo right in the next three flips (not counting for someone else winning before they do)
If the first two flips are HH, player A will now have a 50/50 chance of winning but player B, will be right behind him, if the next coin is T, player B now only needs a H to win in 4 flips.
However, if my maths (as I'm British) is correct, player D has the best chance overall since after an initial T they can never be knocked back to the start. Obviously, the fact that some winning combos overlap complicates the matter but I'm pretty sure this trick works.

[tychowozniaki]

I totally didn't understand that they were flipping coins on their own.
If it was just one coin (which I thought the whole video through) this wouldn't matter, since after the coin showed heads once it's a 50 50 chance.
You explained this very poorly.

[harinathan]

Once its clear there are separate coins, this can be seen in a much easier way. In essence, both are waiting for the first head on their coin. If you are looking for HT, you can keep getting heads until you get the tail you want, and then you are done. However, if you are looking for HH, if you don't get the second head you want, you need to start over. The state diagrams at 1:23 show this as well, for HT, once "get out" of the start state, you are never forced back in. However, for HH, once you "get out" of the start state, you have a 50% chance of being forced back in. Of course, this reasoning doesn't get you the exact number of expected flips, but it may be an easier way to see why its not equal.

[mursalinhabib]

The people who are saying this is wrong probably missed that there were *two* coins involved: one for Wilbur and one for Orville.

It is true that the game is fair if there were only one coin involved. But this is not the case if Wilbur and Orville both have separate coins.

The expected number of coin flips required to get two heads in a row is higher than the expected number of flips required to get a head followed by a tail. So, in this sense, two heads in a row is more rare.

[IDMYM]

To all those who are saying it is misleading and made us assume that there was 1 coin, listen carefully here 0:24
_" so that Orville would win as soon as two heads showed up in a row on _*_his_*_ coin, _
_and Wilbur would win as soon as heads was immediately followed by tails on _*_his?_*_ "_