The coin flip conundrum - Po-Shen Loh
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Date: 2020-08-22
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Comments and reviews: 10
education
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
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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
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Renzo
Nice video but the problem should be stated more precisely: Po-Sheh clarified each player tosses their own coin. But even like so there are two ways the game could be ruled: if player one gets head-head (potential win) but player 2 gets head-tail at the same time this is a draw, so what you do? continue tossing using the previous tosses, or start over?
I calculate the probabilities and it goes like this:
prob. palyer head-head wins =39/121=32. 2. % (without having a draw before)
prob player head-tail wins=65/121 (without having a draw before.
rob to get a draw = 17/121
So if you start over when a draw:
prob. HH player wins= 39/104
prob HT player wins = 65/104
but if you continue tossing after first, second, or any number of draws (tougher calculation this one):
prob HH player wins = 47/114
prob HT player wins = 67/114
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Nice video but the problem should be stated more precisely: Po-Sheh clarified each player tosses their own coin. But even like so there are two ways the game could be ruled: if player one gets head-head (potential win) but player 2 gets head-tail at the same time this is a draw, so what you do? continue tossing using the previous tosses, or start over?
I calculate the probabilities and it goes like this:
prob. palyer head-head wins =39/121=32. 2. % (without having a draw before)
prob player head-tail wins=65/121 (without having a draw before.
rob to get a draw = 17/121
So if you start over when a draw:
prob. HH player wins= 39/104
prob HT player wins = 65/104
but if you continue tossing after first, second, or any number of draws (tougher calculation this one):
prob HH player wins = 47/114
prob HT player wins = 67/114
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Jason
The way this problem is worded is misleading. They should emphasize that the condition has changed from flipping one coin to each of the two brothers flipping a coin of their own(I feel there was an intent to mislead by de-emphasize this change. And in this case, it should also be made clear that the brother who flips his winning pattern in a fewer number of flips wins. Not that most of us would be confused into thinking that the faster flipper would have a better chance to win; rather, it would have clarified the condition for most of us that they are flipping separately and that when one brother flips the winning pattern of the other would not conclude the game. I feel this is as much a logic puzzle in rhetoric as it is in probability. Tricky.
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The way this problem is worded is misleading. They should emphasize that the condition has changed from flipping one coin to each of the two brothers flipping a coin of their own(I feel there was an intent to mislead by de-emphasize this change. And in this case, it should also be made clear that the brother who flips his winning pattern in a fewer number of flips wins. Not that most of us would be confused into thinking that the faster flipper would have a better chance to win; rather, it would have clarified the condition for most of us that they are flipping separately and that when one brother flips the winning pattern of the other would not conclude the game. I feel this is as much a logic puzzle in rhetoric as it is in probability. Tricky.
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Digital
To simplify this even further, any time you need the same result twice in a row it will be less probable. This is magnified with a sample size of two (heads and tails. The first flip is always a 50% shot for both players. So is the second flip. But when you fail on the second flip is where it changes. P1 needs H-H, P2 needs H-T. Let's say both get H on flip 1. Then say both fail flip 2. P1 gets T, P2 gets H. P2 is still only one flip away from H-T, because their last flip was an H. P1 must get H-H. Their prior flip was a T, so they need a minimum of two flips to get H-H, which the probability is now at 25% to P2's 50%.
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To simplify this even further, any time you need the same result twice in a row it will be less probable. This is magnified with a sample size of two (heads and tails. The first flip is always a 50% shot for both players. So is the second flip. But when you fail on the second flip is where it changes. P1 needs H-H, P2 needs H-T. Let's say both get H on flip 1. Then say both fail flip 2. P1 gets T, P2 gets H. P2 is still only one flip away from H-T, because their last flip was an H. P1 must get H-H. Their prior flip was a T, so they need a minimum of two flips to get H-H, which the probability is now at 25% to P2's 50%.
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Balendula
Why is this so hard for people to understand? The probability that you will get 2 of the same outcomes in the row is smaller than 2 different outcomes. The probability gets smaller the more times in a row you are trying to get the same outcome. It's very simple. The video made the mistake of using a coin flip as an example, as you conflate the 50/50 with the two flips. A better example would be trying to flip heads 5 times in a row or something like that.
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Why is this so hard for people to understand? The probability that you will get 2 of the same outcomes in the row is smaller than 2 different outcomes. The probability gets smaller the more times in a row you are trying to get the same outcome. It's very simple. The video made the mistake of using a coin flip as an example, as you conflate the 50/50 with the two flips. A better example would be trying to flip heads 5 times in a row or something like that.
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Jordan
If they're only going until the first instance of HH or HT, then the resulting string will be T. TH (with any non-negative number of tails) followed by either a head or a tail. For the first occurrence both will be determined by this single last coin flip making the probability still 50%. For a total count in a fixed number of flips it would be different or even if they were looking for TH instead of HT.
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If they're only going until the first instance of HH or HT, then the resulting string will be T. TH (with any non-negative number of tails) followed by either a head or a tail. For the first occurrence both will be determined by this single last coin flip making the probability still 50%. For a total count in a fixed number of flips it would be different or even if they were looking for TH instead of HT.
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Alan
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.
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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.
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Doctor
The definition of Insanity is to do the same thing over and over again and expect a diffrent resault. Albert Einstein. What if you flip a Coin three times and bet on a Heads Tail heads sequnce? Then a Tails heads heads flip next turn? Would you be Crazy? Not to Double down?
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The definition of Insanity is to do the same thing over and over again and expect a diffrent resault. Albert Einstein. What if you flip a Coin three times and bet on a Heads Tail heads sequnce? Then a Tails heads heads flip next turn? Would you be Crazy? Not to Double down?
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education
Tldr
When trying for HT, if you fail on the second throw and get HH, you're already halfway to completing the next HT
If you're trying for HH and get HT, you haven't gotten started on the next HH at all
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Tldr
When trying for HT, if you fail on the second throw and get HH, you're already halfway to completing the next HT
If you're trying for HH and get HT, you haven't gotten started on the next HH at all
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Andrew
Actually, they have the same chance. The first for both of them is heads, so we can cancel it out. Then it's either heads or tails which means they still have a equal chance of testing the flight
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Actually, they have the same chance. The first for both of them is heads, so we can cancel it out. Then it's either heads or tails which means they still have a equal chance of testing the flight
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