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Definitely. If your goal is to maximize the number of correct guesses across all questionings, then always answer 1/3. If your goal is to guess the current co

by psyklic 18y ago

Definitely. If your goal is to maximize the number of correct guesses across all questionings, then always answer 1/3. If your goal is to guess the current coin flip as accurately as possible, then always answer 1/2. (I interpret the question as this.)

emmett 18y ago
Try thinking about it this way instead: If the coin comes up tails, you will be asked 1,000,000 times in a row. If the coin comes up heads, you have a 1/1,000,000 chance of being asked. I wake you up. If you had to guess if the coin was heads or tails, what would you say?
- GavinB 18y ago
  The reason that people are confused is because of the word "probability." I guess they don't realize that probability is simply a tool that tells you what to guess.
- psyklic 18y ago
  You are proposing the 1/3 argument -- by answering tails, she will get it right more often across trials. But, 1/2 of the time she will be awoken once, and 1/2 she will be awoken 1,000,000 times. So, let's suppose she is awoken. There are 1,000,001 possible awakenings. The probability of her being in the first of these is 1/2 (Heads). The probability of her being in any particular other is 1/2*(1/1,000,000) (Tails). Hence, she should answer 1/2 if she wants to guess the current coin flip correctly, since the chance of the coin being Heads on any arbitrary awakening is 1/2.
  - emmett 18y ago
    If we actually performed that experiment, and you were to be killed if you guessed wrong, what would you answer to the question "Is the coin heads or tails?"? Then ask yourself, how afraid of dying would you be? (In particular, the reason why the probabilities don't work out that way is that on any given awakening, you have a 1 in 1,000,001 chance of it being heads, and a 1,000,000 in 1,000,001 chance of it being tails). If these previous arguments are not convincing, consider this (equivalent) alternative: what if you were woken up once on tails, but only 1 in 1,000,000 times on heads? What would you give as your credence that the coin came up tails then?
- trevelyan 18y ago
  There is a 50% chance of the coin landing on Heads, and a 50% of it landing on Tails. The events in question are not independent. You have a 50% chance of being woken once, and a 50% chance of being woken 1,000,000 times. Therefore the correct answer is 50%.
  - GavinB 18y ago
    I guess the question is whether being wrong 1,000,000 times is equivalent to being wrong 1 time. Sadly, the problem doesn't define that.
GavinB 18y ago
If you're being asked, then you know that a bet of Heads will be right 1/3 of the time. So the probability from your perspective is 1/3. It doesn't make a different if you're only doing it once or a million times. The whole point of probability is to figure out what would happen if you did it a million times so that you use the optimal strategy when it happens only once.
Tichy 18y ago
I don't understand what you are trying to say. Why shoul it make a difference if you want to maximize correct guesses across all questionings, or across one questioning? I don't think it does - the answer is always 1/3.

by psyklic 18y ago

emmett 18y ago

Try thinking about it this way instead: If the coin comes up tails, you will be asked 1,000,000 times in a row. If the coin comes up heads, you have a 1/1,000,000 chance of being asked. I wake you up. If you had to guess if the coin was heads or tails, what would you say?

GavinB 18y ago
The reason that people are confused is because of the word "probability." I guess they don't realize that probability is simply a tool that tells you what to guess.
psyklic 18y ago
You are proposing the 1/3 argument -- by answering tails, she will get it right more often across trials. But, 1/2 of the time she will be awoken once, and 1/2 she will be awoken 1,000,000 times. So, let's suppose she is awoken. There are 1,000,001 possible awakenings. The probability of her being in the first of these is 1/2 (Heads). The probability of her being in any particular other is 1/2*(1/1,000,000) (Tails). Hence, she should answer 1/2 if she wants to guess the current coin flip correctly, since the chance of the coin being Heads on any arbitrary awakening is 1/2.
- emmett 18y ago
  If we actually performed that experiment, and you were to be killed if you guessed wrong, what would you answer to the question "Is the coin heads or tails?"? Then ask yourself, how afraid of dying would you be? (In particular, the reason why the probabilities don't work out that way is that on any given awakening, you have a 1 in 1,000,001 chance of it being heads, and a 1,000,000 in 1,000,001 chance of it being tails). If these previous arguments are not convincing, consider this (equivalent) alternative: what if you were woken up once on tails, but only 1 in 1,000,000 times on heads? What would you give as your credence that the coin came up tails then?
trevelyan 18y ago
There is a 50% chance of the coin landing on Heads, and a 50% of it landing on Tails. The events in question are not independent. You have a 50% chance of being woken once, and a 50% chance of being woken 1,000,000 times. Therefore the correct answer is 50%.
- GavinB 18y ago
  I guess the question is whether being wrong 1,000,000 times is equivalent to being wrong 1 time. Sadly, the problem doesn't define that.

GavinB 18y ago

If you're being asked, then you know that a bet of Heads will be right 1/3 of the time. So the probability from your perspective is 1/3. It doesn't make a different if you're only doing it once or a million times. The whole point of probability is to figure out what would happen if you did it a million times so that you use the optimal strategy when it happens only once.

Tichy 18y ago

I don't understand what you are trying to say. Why shoul it make a difference if you want to maximize correct guesses across all questionings, or across one questioning? I don't think it does - the answer is always 1/3.