Always 50/50 chance?

[drkitten]

- If the probabilities are 50/50, and the events are completely independent or behave independently, then it really does not matter what strategy you use in any way.

And, as Major Joseph Vernam and Claude Shannon showed, you've just reinvented the one-time pad method of cryptography.

- If the probabilities favour outcome X vs. outcome Y, no matter how much better, always bet on the favorite X (again, if IID or pseudo-IID events). (I.E. forget everything I said in red above. The increased variance does not make up for the decreased base accuracy, so always bet the favorites.)

Yup.

So here's a question. You bet randomly Heads/Tails for 30 times, and each of those 30 times the toss is Tails. Do you change your strategy now?

Am I permitted to question the fairness of the coin (flips)?

If the coin is irrebutably presumed to be fair, then, no. I just got realliy unlucky. If the presumption that the coin is fair is rebuttable, I suggest that thirty consecutive tails -- for a net probability of approximately a billion-to-one against -- far exceeds any reasonable alpha cutoff, and gives me grounds to reject the fairness of the coin.

 

[Dave1001]

Only if you assume an infinite population. I think

I think that solves my concern. A psychic operates in a temporal frame that limits the total birth or coin toss predictions they can make.

So I think a very productive psychic in these areas can be expected to be about maximally wrong if they choose to predict one particular gender or one particular coin side 75% of the time. Maybe this has to do with expected 1:1 variance for these non-infinite populations?

I haven't yet taken a probability or statistics course yet, nor have I read deeply in these areas, so a clear explanation of all this would be appreciated. In the mean time I'm off to see what wikipedia and google has to say.

 

[Dave1001]

This is amazing, given the number of things still decided by coin toss.

From wikipedia probability section.

"The Not So Random Coin Toss, Mathematicians Say Slight but Real Bias Toward Heads". NPR.

 

[RecoveringYuppy]

So I think a very productive psychic in these areas can be expected to be about maximally wrong if they choose to predict one particular gender or one particular coin side 75% of the time.

No. My fault for first bringing up this misunderstanding.

Let's go with coin tosses and assume a perfect 50/50 "fairness". The coin toss is completely independent of how you make your guess as to what it's going to come up. No matter what you guess or how you arrive at it. As long as you guess either heads or tails you have a 50/50 chance of being right. The coin doesn't care a bit about how you arrived at that decision.

And "population" size doesn't matter either because the coin doesn't know how many times you're going to toss it.

 

[rtalman]

I was intrigued by all of the talk about pattern guessing vs. random choice, so I made a quick Excel worksheet to test the correct % of guesses based on choosing GGGB, All Girl, All Boy, and random guessing.

Based on 1000 "births"
Trial 1
GGGB repeated 467 correct
G Only 483 correct
B Only 517 correct
Random 491 correct

Trial 2
GGGB repeated 492 correct
G Only 506 correct
B only 494 correct
Random 488 correct

Trial 3
GGGB repeated 494 correct
G Only 482 correct
B Only 518 correct
Random 495 correct

Trial 4
GGGB repeated 497 correct
G Only 511 correct
B only 489 correct
Random 495 correct

Trial 5
GGGB repeated 502 correct
G Only 514 correct
B Only 486 correct
Random 488 correct

 

[Dave1001]

No. My fault for first bringing up this misunderstanding.

Let's go with coin tosses and assume a perfect 50/50 "fairness". The coin toss is completely independent of how you make your guess as to what it's going to come up. No matter what you guess or how you arrive at it. As long as you guess either heads or tails you have a 50/50 chance of being right. The coin doesn't care a bit about how you arrived at that decision.

And "population" size doesn't matter either because the coin doesn't know how many times you're going to toss it.

I still think the psychic will likely be about maximally wrong guessing one gender or one coin side 75% of the time with a non-infinite population, because it seems to me they're making an incorrect guess on variance. Something that I think emerges out of multiple toin cosses/births rather than a single toss or birth.

My continuing invitation for someone with greater knowledge on this topic to fill in the holes in our discussion remains open.

 

[Dave1001]

I was intrigued by all of the talk about pattern guessing vs. random choice, so I made a quick Excel worksheet to test the correct % of guesses based on choosing GGGB, All Girl, All Boy, and random guessing.

Based on 1000 "births"
Trial 1
GGGB repeated 467 correct
G Only 483 correct
B Only 517 correct
Random 491 correct

Trial 2
GGGB repeated 492 correct
G Only 506 correct
B only 494 correct
Random 488 correct

Trial 3
GGGB repeated 494 correct
G Only 482 correct
B Only 518 correct
Random 495 correct

Trial 4
GGGB repeated 497 correct
G Only 511 correct
B only 489 correct
Random 495 correct

Trial 5
GGGB repeated 502 correct
G Only 514 correct
B Only 486 correct
Random 488 correct

Definitely nominated. Now I'll take a look at the numbers.

 

[Dave1001]

okay, I took a look. I noticed GGGB was NEVER correct the most often, if I skimmed the numbers correctly. I think if we add up all the results, the spread will be striking, with all B, all G, and random becoming closer to 500 and each other and GGGB becomig noticeably lower than 500 and the rest.

Let's see if I have egg on my face.

 

[drkitten]

I still think the psychic will likely be about maximally wrong guessing one gender or one coin side 75% of the time with a non-infinite population, because it seems to me they're making an incorrect guess on variance. Something that I think emerges out of multiple toin cosses/births rather than a single toss or birth.

No. Assuming that births/coin flips are independent, (which is probably a valid assumption, but I don't think anyone has made it explicit yet), then the probability distribution over any finite population will be controlled by the finite binomial distribution.

Simple algebra at this point will confirm that if the probability of a girl/heads is exactly 0.5, then it doesn't matter what you pick, and if the probability is > 0.5, then your correctness will go up with the percentage of times you pick girl/heads, achieving a maximum at 100%.

 

[Dave1001]

No. Assuming that births/coin flips are independent, (which is probably a valid assumption, but I don't think anyone has made it explcit yet), then the probability distribution over any finite population will be controlled by the finite binomial distribution.

Simple algebra at this point will confirm that if the probability of a girl/heads is exactly 0.5, then it doesn't matter what you pick, and if the probability is > 0.5, then your correctness will go up with the percentage of times you pick girl/heads, achieving a maximum at 100%.

okay, that makes perfect sense and a great explanation.
One area of clarification: Does that mean that maximal wrongness occurs in picking a specific gender or coin side about 50% of the time? (I say about becaus probability of these particular examples is not exactly 0.5). If so, then I understand your explanation.

 

[Baron Samedi]

Am I permitted to question the fairness of the coin (flips)?

If the coin is irrebutably presumed to be fair, then, no. I just got realliy unlucky. If the presumption that the coin is fair is rebuttable, I suggest that thirty consecutive tails -- for a net probability of approximately a billion-to-one against -- far exceeds any reasonable alpha cutoff, and gives me grounds to reject the fairness of the coin.

Hrm... so you wouldn't start betting "Heads" more often now, because "Heads" are due and as we all know, the coin just HAS to be 50/50?

Don't worry, my answer to this question was yours as well... run, run fast and far... unless you're the one tossing the coin. Then please stay.

No. My fault for first bringing up this misunderstanding.

Let's go with coin tosses and assume a perfect 50/50 "fairness". The coin toss is completely independent of how you make your guess as to what it's going to come up. No matter what you guess or how you arrive at it. As long as you guess either heads or tails you have a 50/50 chance of being right. The coin doesn't care a bit about how you arrived at that decision.

And "population" size doesn't matter either because the coin doesn't know how many times you're going to toss it.

I only brought up population size to stress that even though the events may not be independend, that if N is big enough, than you can assume independence without being off by too much. If I draw 5 Hearts from a shuffled deck of cards, the probability that the next card is also Hearts is 8/47. If I draw 5 Hearts from a set of 200 shuffled decks, the probability that the next card is also a Hearts is very very very close to 1/4, so you may as well just round off and be done with it.

 

[Baron Samedi]

I was intrigued by all of the talk about pattern guessing vs. random choice, so I made a quick Excel worksheet to test the correct % of guesses based on choosing GGGB, All Girl, All Boy, and random guessing.

Based on 1000 "births"
Trial 1
GGGB repeated 467 correct
G Only 483 correct
B Only 517 correct
Random 491 correct

Trial 2
GGGB repeated 492 correct
G Only 506 correct
B only 494 correct
Random 488 correct

Trial 3
GGGB repeated 494 correct
G Only 482 correct
B Only 518 correct
Random 495 correct

Trial 4
GGGB repeated 497 correct
G Only 511 correct
B only 489 correct
Random 495 correct

Trial 5
GGGB repeated 502 correct
G Only 514 correct
B Only 486 correct
Random 488 correct

Each of these numbers should follow the same distribution, with a mean of 500 with 1 s.d. of +/- 15.8. They jive.

Let me try with a few thousand reps. (I love SAS, whee)

 

[RecoveringYuppy]

I only brought up population size to stress that even though the events may not be independend, that if N is big enough, than you can assume independence without being off by too much. If I draw 5 Hearts from a shuffled deck of cards, the probability that the next card is also Hearts is 8/47.

Agreed. I think I was addressing someone elses use of "population". My point being that the coin doesn't know nor care how many times it has been/will be flipped. You're card dealing example is correct: To maintain the independence of each draw you need an infinite deck. Any finite deck is going to be dependent on previous draws (Card counting in blackjack).

 

[Dave1001]

okay, I took a look. I noticed GGGB was NEVER correct the most often, if I skimmed the numbers correctly. I think if we add up all the results, the spread will be striking, with all B, all G, and random becoming closer to 500 and each other and GGGB becomig noticeably lower than 500 and the rest.

Let's see if I have egg on my face.

According to DrKitten's post, should likely correctness rankings should be:
1. all B & all G (tie)
2. GGGB
3. random

The question is why? I think the answer is the possibility of being completely wrong is reduced as random variance of a given possibility is reduced from 100%.

So, let's say the real outcomes are something like

GGGBGBBGBBGGBGGGGBGBGBBBBBGGBB

with completely random variance I maximize my chances of getting it completely wrong.

BBBGBGG...

The more consistently I use either G or B, the more of those I'm not going to approach being completely incorrect in this way.

have to go, (evidence lecture over) will flesh out later.

 

[drkitten]

One area of clarification: Does that mean that maximal wrongness occurs in picking a specific gender or coin side about 50% of the time? (I say about becaus probability of these particular examples is not exactly 0.5). If so, then I understand your explanation.

I'm afraid I don't quite understand your question.

If the probability of getting a girl/heads is exactly 0.50 and the births/flips are independent, then it doesn't matter what you guess. You can always guess "girl," never guess "girl," guess "girl" only on trials that are prime numbers, ... and you will always be (in the long run) about 50% correct. The only way you can substantially lower your accuracy rate is by making guesses such as "the coin will neither land heads nor tails, but will instead hover at the ceiling making 'beep beep' noises like the Road Runner."

If the probability of getting a girl is greater than 50% (and still assuming independence), then your best strategy is to always guess "girl," and your worst strategy is to never guess girl. In particular, if the probability of getting a girl is (50+x)%, then you will be (50+x)% correct if you always guess "girl," (50-x)% correct if you never guess "girl," and if you choose girl with probability y, then your overall corectness will be [(50+x)*y] + [(50-x)*(1-y)]. (Note that as a probability, y ranges from 0 to 1, inclusive).

IF I've done my algebra right, that final expression is equivalent to 50 -x + 2xy. Since x is positive, 2xy is a positive quantity and increases linearly with y. So to maximize its value, set y to be as large as possible, and to minimize it, set y to be as small as possible. (Note also that at the extrema, y=0 or y=1, this equation reduces to the previous special cases, as expected).

 

[drkitten]

According to DrKitten's post, should likely correctness rankings should be:
1. all B & all G (tie)
2. GGGB
3. random

Er, no. According to my post, all four strategies should tie. See previous post.

 

[RecoveringYuppy]

One area of clarification: Does that mean that maximal wrongness occurs in picking a specific gender or coin side about 50% of the time? (I say about becaus probability of these particular examples is not exactly 0.5). If so, then I understand your explanation.

If I understand your question:

If you have a situation where there is a 75/25 split you achieve "maximal wrongness" by simply always guessing the opposite of whatever has the 75% probability. Beyond that the reasoning for this case is not much different from the 50/50 case. No matter how you make your guess, you have 75/25 odds. The one difference is that if the guesser knows what the odds are they have two choices which they can choose between: They can be 75% right or 75% wrong in the long run*. Either of those two cases indicates "guessing", without real foreknowledge of the outcome.

* Now ask what happens if the guesser alternates between those two strategies.

 

[Baron Samedi]

Here we go. 10,000 simulations (lets see if I can format this nicely):

Variable Q1 Mean Median Q3 Std Dev
total_right_allheads 489 500.099 500 511 15.7314318
total_right_alltails 489 499.901 500 511 15.7314318
total_right_HHHT 489 499.9916 500 511 15.7228512
total_right_random 490 499.9442 500 510 15.6885659

So they're all the same.... within error

 

[Dave1001]

I'm afraid I don't quite understand your question.

If the probability of getting a girl/heads is exactly 0.50 and the births/flips are independent, then it doesn't matter what you guess. You can always guess "girl," never guess "girl," guess "girl" only on trials that are prime numbers, ... and you will always be (in the long run) about 50% correct. The only way you can substantially lower your accuracy rate is by making guesses such as "the coin will neither land heads nor tails, but will instead hover at the ceiling making 'beep beep' noises like the Road Runner."

If the probability of getting a girl is greater than 50% (and still assuming independence), then your best strategy is to always guess "girl," and your worst strategy is to never guess girl. In particular, if the probability of getting a girl is (50+x)%, then you will be (50+x)% correct if you always guess "girl," (50-x)% correct if you never guess "girl," and if you choose girl with probability y, then your overall corectness will be [(50+x)*y] + [(50-x)*(1-y)]. (Note that as a probability, y ranges from 0 to 1, inclusive).

IF I've done my algebra right, that final expression is equivalent to 50 -x + 2xy. Since x is positive, 2xy is a positive quantity and increases linearly with y. So to maximize its value, set y to be as large as possible, and to minimize it, set y to be as small as possible. (Note also that at the extrema, y=0 or y=1, this equation reduces to the previous special cases, as expected).

okay, this is helping. Although I suspect it's a great waste of collective cognitive resources for me to learn basic probability in this way.

ETA: This is a particularly helpful clarification: "If the probability of getting a girl is greater than 50% (and still assuming independence), then your best strategy is to always guess "girl," and your worst strategy is to never guess girl." I misread this concept in earlier posts to apply when the probability of getting a girl is 50%. It's intuitive that this applies when the probability of getting a girl is greater than 50%. Nominated.

 

[drkitten]

Here we go. 10,000 simulations (lets see if I can format this nicely):

Variable Q1 Mean Median Q3 Std Dev
total_right_allheads 489 500.099 500 511 15.7314318
total_right_alltails 489 499.901 500 511 15.7314318
total_right_HHHT 489 499.9916 500 511 15.7228512
total_right_random 490 499.9442 500 510 15.6885659

So they're all the same.... within error

Monte Carlo simulation for the win.... Thank you very much, my Lord. (Did I get the address right for a Baron? I rarely hang out with such exalted persons....)