Penney’s Game – Numberphile

Penney’s Game – Numberphile


I’m gonna show you a little bit of a scam, or a trick, that you can try out on people. So, what I’m going to ask you to do, is to pick a sequence of three coin tosses. So, you know, heads-tails-heads, or tails-tails-tails. Brady: “Tails-tails-heads.” Okay. So, Brady’s picking tails-tails-heads. And I will pick heads-tails-tails. And the game is: we’re gonna toss this coin and see who gets that first. So, let’s see who wins. We’ll do a best of 5. Okay? Something like that. Okay, so, let’s try game Number 1. All right, here we go. So, number 1… All right, let’s write down this sequence. Heads. Okay, that’s pretty good actually; that’s good for me. Yeah, it’s good for me, not good for you. All right, heads again. Heads again. Ooh, look at this. We might have to do a different video about how many heads we can get in a row. Oh look, tails. Alright Oooh, tails. That is a win for me. Excellent. A James win, brilliant. Alright, well, best of 5? Alright, Brady’s got a tail. Come on. Brady: “Ah, now, this is good for me.” Yes! That’s a tail. Brady: “Yes!” – And… It’s a tail! Oh no you’re okay though You’re still, you’re still in the lead Yeah, happy with that aren’t you? Alright, oooh Brady: “That’s good for me.” Tails, tails Brady: “In fact, I’m definitely gonna win this next one” Soon as the sequence breaks, yeah, you’re right. Well observed, Brady. Brady does win that, he had to win that after all those tails coming up. Okay, Brady: “So, It is bad to choose any two consecutives as you start.” It is bad to choose. Brady! Brady: “I realize what I did wrong.” Ahhh cmon cmon cmon you can still win this Have you given up? Brady: “I’ve realized my mistake.” You’ve realized what a mistake You’re less likely to win than I am. Brady: “So, what would you have done if I had not been stupid and not given consecutives as my…” Whatever sequence that you pick I can pick a sequence that is more likely to appear before yours. Brady: “What if I pick the most likely of them all?” Exactly, what if you pick the most likely of them all? It goes round in a circle like rock, paper, scissors Let’s say that Brady picks whatever sequence he does pick I’m going to show you what sequence I should pick so that I can beat him. Right, or at least with a better chance of beating him. I’m going to show you the cycle of winning. Let’s do that. Right. So let’s say Brady picks heads-heads-heads, right? Three heads in a row. Then what I should pick is tail-heads-heads. And tail-heads-heads will beat heads-heads-heads. If Brady had picked tails-heads-heads, then I should choose tails-tails-heads. Because that is going to beat Brady. If Brady had picked tails-tails-heads, then I should pick heads-tails-tails, that will beat Brady. Incidentally, if he had picked, instead, tails-heads-tails, you pick this one here, tails-tails-heads. So it’s not quite a circle, this. It has some spikes coming off the side of it. If Brady had picked tails-tails-tails, you want to choose heads-tails-tails. If Brady had picked heads-tails-tails, then the best choice is heads-heads-tails. If you had picked heads-tails-heads, I would again pick heads-heads-tails. If you had chosen heads-heads-tails, Brady, I would have picked tails-heads-heads. So we have this little circle, but you can see the spikes coming off it as well. So to read this, Brady’s choice would be here, at the pointy end of the arrow. My choice, the better choice, would be here, so the arrow going towards Brady’s choice there. There is a way to help you to remember this cycle. The way you do it is, when your opponent picks a sequence, let’s say, tails-heads-heads, what you should do is, in your mind, copy the middle one, so make a copy of that coin, so I’ve got another heads there, I’m gonna put it at the front, and flip it over. So it would become a tails. And then my choice would be tails-tails-heads. That’s the winning choice. That little way of remembering it is better than just remembering that cycle. And that will work. That’s your best choice to go for. I should show you the probabilities for each choice. If Brady picked heads-heads-heads, I would pick tails-heads-heads. And the chance of beating you is actually pretty good. It’s seven eighths. I can tell you that’s round about 87.5 percent. So it’s really likely. I’m really likely to win the game. That’s why I did best of five, though, just in case probability let me down. This probability here, if I wanted to work tails-heads-heads beating heads-heads-tails, that happens with a chance of three quarters, so it’s about 75 percent, really big probability. If you wanted to do this one here, heads-heads-tails beating heads-tails-heads, that happens with a chance of two thirds, so that’s, what’s that, about 67 percent. And then, actually the others are similar. This probability here is two thirds, this one here is seven eighths, this one here is three quarters. There’s a symmetry in this. So you are far more likely to win. Excellent. Try it out on your enemies, right. So you can beat them. This is insane, because you think, well, surely there’s a best choice. There’s a best choice, and then, all the others are worse than that. And you can’t beat the best choice. It works in such a strange way that it makes this cycle of probabilities instead. Just to show you where these come from. These are some of the easy ones, okay? I’m gonna show you where some of these come from. Well, look, this is an easy one, look at this. I said this was seven eighths. Yeah, that’s actually an easy one to spot, because, well, you could get heads-heads-heads straightaway, which happens one eighth of the time. But if you don’t get heads-heads-heads straightaway, if you’ve got a sequence, and somewhere in the sequence it’s heads-heads-heads, let’s say this is the first appearance, then it has to be preceded by a tail. If it’s not, if it was preceded by a head, then it wouldn’t be the first heads-heads-heads in the sequence. It has to be preceded by a tail. Which means it has to have tails-heads-heads coming up before it unless you get the three heads in a row straightaway. So, yeah, you’re going to lose if you pick heads-heads-heads, most of the time. Yeah, so we go back to our game, what Brady chose. He chose tails-tails-heads, it wasn’t the best choice. Brady, if you picked tails-tails-heads, I used my little algorithm, I know what I’m supposed to choose, heads-tails-tails, and I can beat you, appear before yours does, with a chance of 75 percent. Which is actually the second best thing there. Really bad choice, Brady. Sorry about that. Well I might not be making the best coin choices, but I have been learning a lot of new mathematical tricks at TheGreatCoursesPlus. This is a great online resource for anyone who wants to learn anything. Become smarter about things from cookery to quantum mechanics. I think numberphile fans, in particular, might be interested by some of the mathematical offerings. They’re really extensive, and right up the alley of the sort of people who watch these videos. There’s lots about games, puzzles, probability. I’ve really been enjoying, just today, some of the videos about probability. I wish I’d watched them before recording that video with James. If you’d like to find out more, go to TheGreatCoursesPlus dot com slash numberphile. Have a look what’s there, and if you like it you can actually sign up for a one-month free trial. That’s one month’s access to 7000 plus videos, all taught by leading experts in these fields. That address again, TheGreatCoursesPlus.com/numberphile. Give it a look. If you like these videos here on numberphile, I think you might really like what you find there. And cheers to the GreatCoursesPlus for supporting this video. This game is called Penney’s Game, or penny ante, which I think is a kind of a pun. Actually, the strange thing is, Penney doesn’t refer to coins and pennies, and flipping coins, It’s actually the name of guy who came up with the game. He was called Penney. It’s kind of one of those situations when you have a baker called Mr. Baker, Or a mathematician called Dr. Sexy or something like that.


100 thoughts on “Penney’s Game – Numberphile

  1. Hey but who would have the advantage if there were four players and they chose the four combinations part of the circle (THH – HHT – HTT – TTH) ?

  2. intersesting Video.
    But i still have one question.
    what if I choose the Opposition Sequenz to the Sequenz of my opponent?
    (For example they choose htt and I choose thh)

  3. supposing that the player picks TTH like Brady has, how do the chances of winning between picking HTT (as recommended) and picking TTT differ?

  4. Just taking a spin on this 😉
    Let's start from say HTT – If it is 67% more likely to get HHT than to get HTT and then 75 % more likely to get THH instead but then 67% more likely to get TTH and then still 75% more likely to get HTT :^)
    Then it could be said that it's 25,25% (67% x 75% x 67% x 75%) more likely to get HTT than to get that same HTT 😀

  5. Do I have to count all probabilities in this circle to understand why it is true?

    It's really counterintuitive.

  6. according to the trick to remember what to choose, if someone chooses THT you should pick TTT. Can anyone explain?
    (flip the H – to T – and put it in front)

  7. Can this be extended to 4 or 5 coin flips to give you even better odds?
    And is there an easy rule to remember what you should choose?
    Intuition tells me that you flip second to last coin, and put it up front.

  8. Now does this extend? as in what would you say if you picked a sequence of 2,4,5, tec? (obviously, 3's already covered in this video) that would be an interesting video, and perhaps doing a numberphile2 video on the logic behind the new choices and an extendable fassion

  9. 7:00 is it actually possible to complete the outer paths of the winning graph with the same rule? If so, wouldn't that thing be isomorphic to the dihedral group D4?

  10. Probabilities are real only if no sequence out of two choices gets it right in the first 3 tosses. That is, each of the 8 options has a 12,5 % propabilitty to appear in the very first three tosses. This is crazy, to believe that luck has a plan. If 8 people are playing, every game will have to stop at the third coin flip because a sequence of the 8 will definetely appear.

  11. Great video. Another way to remember this patter is to note the first two picks. If they are identical (HH or TT), you inverse your first pick and repeat those two picks (THH or HTT). For an alternate sequence (TH or HT), follow the one-one-two pattern (TTH or HHT).

  12. THH and HTT are the best options for the one who picks first. Because, at the worst, you lose with a 67%, but at the best, you win with an 87.5%.

  13. is it possible to use this strategy during a game of darts? each player would choose a triple color sequence and whenever a dart hits the target,they record the space's color…

  14. This makes perfect sense when you have your opponent pick HHH or TTT. But, what if he picks HTH. Why is HHT better than THT?

    In other words, if my opponent picks XYZ where all three are not the same, Why should TXY and HXY not be the same probability?

  15. 8 elements have 28 groups of 2. In your diagram you have measured only 8 relations. Are ALL the rest related by a 50% chance?. (Some of them, like TTT vs HHH or THT vs HTH are clear, but, what about the others?)

  16. Why does this happen? Shouldn't the probability be 50/50? Are the coins uneven and so this weirdness happens? That obviously leads into>Will ideal coins result in this result or 50/50? Guess I should watch the extra footage…

  17. here's how to fix the game.

    have all players pick theirs in private and reveal right before the game begins.

    this makes strengths and weaknesses happen more randomly, and there is always a chance that players will have random benefits.

    or just force all players to pick one of the 4 cardinal choices (TTT, HHH, THT, HTH) which leaves everyone with an equal probability.

  18. Probability may be the most counter-intuitive branch of mathematics. Similar to the "non-transitive" coin flipping game in this video are "non-transitive dice", covered in another Numberphile video. Then you've got the "Monty Hall Problem", as well as the "Bertrand Paradox", both of which can be found in a quick Internet search. I recall another youtube video about an experiment which indicated that birds, pigeons in particular, seem to have better intuition than people in situations rather like the Monty Hall Problem.

  19. Where did he get the chance of beating another sequence from? How does he know X sequence has a 75% chance of beating Y?

  20. if I use the rule, the sequence to beat HHT is THT instead of THH? seems the rule suggest not to change the last letter meaning a sequence will only be beaten by a sequence ending with the same letter. did I miss anything?

  21. Why is the "flip" of the middle coin necessary when constructing the winning sequence? Shouldn't HTH be just as likely to win against THH as TTH is?

  22. How are the probabilities calculated? The probability of each sequence of three coin tosses is 12.5 percent so how is one more probable than the other?

  23. I would call tossing throwing the coin up in the air to increase the randomness or spinning it on a desk to increase the randomness and to build up tension between "tosses".

  24. Okay. I have a challenge for you, James. Find out a method to create a more probable sequence of length n. Please?

  25. Another thing that makes this game counter-intuitive is that you give your victim first choice on selecting his sequence before you select yours. In many games, the person going first has an advantage, so he thinks he has the edge. In this game, it is just the reverse.

  26. I'm curious to know if a winning strategy could be made if you were playing with an n sided "coin." Could you have an edge if this were played with a 3 sided coin?

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