Always 50/50 chance?

[RecoveringYuppy]

A so-called "medium" who could only get 40% accuracy wouldn't even be as accurate as my coin flips. On the other hand, a medium who could consistently get 60% or better would be impressive -- but consistency would have to be demonstrated.

In many cases being consistently 40% wrong is just as impressive and useful as 60% right. Almost any deviation from 50% would be surprising and indicative of something beyond guessing.

ETA: On second thought, you would have to rule out some of the easy ways to be wrong. For example: if a medium guesses "boy" 3 times out of 4 they've guaranteed a failure rate that deviates from guessing without implying any actual knowledge.

 

[Baron Samedi]

In many cases being consistently 40% wrong is just as impressive and useful as 60% right. Almost any deviation from 50% would be surprising and indicative of something beyond guessing.

ETA: On second thought, you would have to rule out some of the easy ways to be wrong. For example: if a medium guesses "boy" 3 times out of 4 they've guaranteed a failure rate that deviates from guessing without implying any actual knowledge.

Ah, not true. I can guess "boy" 100% of the time and still be right with the same rate of dumb luck. So with your example, the outcomes are:

Guess: Boy -- Actual: Boy -- p=3/8
Guess: Boy -- Actual: Girl -- p=3/8
Guess: Girl -- Actual: Boy -- p=1/8
Guess: Girl -- Actual: Girl -- p=1/8
Total times right = 4/8 = 50%

Now, try this same logic on a die. The roll can be a 6 (p=1/6) or it can be not be a six (p=5/6). You may think that the "dumb luck" strategy would be the user choosing 6 about 1/6th of the time, and picking "other than" 6 about 5/6th of the time. If you use this strategy, you will be correct:

Guess: 6 (p=1/6) Actual: 6 (p=1/6) Total p=1/36
Guess: 6 (p=1/6) Actual:not 6 (p=5/6) Total p=5/36
Guess: not 6 (p=5/6) Actual: 6 (p=1/6) Total p=5/36
Guess: not 6 (p=5/6) Actual: not 6 (p=5/6) Total p=25/36
Total accuracy = 26/36 ~ 72%

The funny thing is that if I guess "not six" all of the time, I will be accurate 83% of the time.

Isn't math phun!

So to go back to your question... yes, a medium who consistantly gets only 40% accuracy on coin tosses is indeedy informative. If you always do the complete opposite of what they say, you now do better than luck. It's like a few of my poker buddies. Some of them lose a heck of a lot more often than they win. And boy, do I love playing poker with them. Heh.

 

[RecoveringYuppy]

Yep. That's what I was getting at with my edit.

 

[Dave1001]

In many cases being consistently 40% wrong is just as impressive and useful as 60% right. Almost any deviation from 50% would be surprising and indicative of something beyond guessing.

ETA: On second thought, you would have to rule out some of the easy ways to be wrong. For example: if a medium guesses "boy" 3 times out of 4 they've guaranteed a failure rate that deviates from guessing without implying any actual knowledge.

I don't think so. I think you were correct before your ETA. A stable guessing pattern of Boy Boy Boy Girl Repeat shouldn't lead do a lower % correct than always guessing Boy or always alternating guesses between Boy & Girl. But I admit it's not intuitive. Hopefully someone can give you a clearer explanation why and whatever logical (or psychological) rule explain why this is both true and non-intuitive.

ETA I think I'm also wrong. But I think the hard limit on deviation from 50% wrong is reached when they choose boy 3 times out of 4. for example, they get closer to being wrong only 50% of the time again if they choose "boy" 7 times out of 8, or if they choose boy 5 times out of 8. I think it has to do with them making an incorrect wager that the human species is keyed to produce boys to girls at a 3:4 ratio, when we're keyed to produce boys to girls at about a 1:1 ratio (I think it's 48:52). The same thing would apply to coin tosses.

An nice, understandable explanation of this phenomenon would be great.

 

[GreedyAlgorithm]

Probability is a tricky thing to define. Unless you're a frequentist (which is perverse and should be illegal), you'll almost certainly have to think about probability as the degree of belief you "should" have that some event will occur.

 

[Baron Samedi]

I don't think so. I think you were correct before your ETA. A stable guessing pattern of Boy Boy Boy Girl Repeat shouldn't lead do a lower % correct than always guessing Boy or always alternating guesses between Boy & Girl. But I admit it's not intuitive. Hopefully someone can give you a clearer explanation why and whatever logical (or psychological) rule explain why this is both true and non-intuitive.

Exactly. You were right before the edit. It the outcomes are 50/50, then the optimal solution is to pick either/or, it doesn't make a difference. So if you pick boys 50% of the time, or 70% of the time, or 3% of the time, you'll still be "accurate", in total, 50% of the time. If one probability is slightly higher than the other, then the optimal solution is to ALWAYS bet on the higher one.

(Note: Current user is perverse and should be illegal)

 

[Dave1001]

Ah, not true. I can guess "boy" 100% of the time and still be right with the same rate of dumb luck. So with your example, the outcomes are:

Guess: Boy -- Actual: Boy -- p=3/8
Guess: Boy -- Actual: Girl -- p=3/8
Guess: Girl -- Actual: Boy -- p=1/8
Guess: Girl -- Actual: Girl -- p=1/8
Total times right = 4/8 = 50%

Now, try this same logic on a die. The roll can be a 6 (p=1/6) or it can be not be a six (p=5/6). You may think that the "dumb luck" strategy would be the user choosing 6 about 1/6th of the time, and picking "other than" 6 about 5/6th of the time. If you use this strategy, you will be correct:

Guess: 6 (p=1/6) Actual: 6 (p=1/6) Total p=1/36
Guess: 6 (p=1/6) Actual:not 6 (p=5/6) Total p=5/36
Guess: not 6 (p=5/6) Actual: 6 (p=1/6) Total p=5/36
Guess: not 6 (p=5/6) Actual: not 6 (p=5/6) Total p=25/36
Total accuracy = 26/36 ~ 72%

The funny thing is that if I guess "not six" all of the time, I will be accurate 83% of the time.

Isn't math phun!

So to go back to your question... yes, a medium who consistantly gets only 40% accuracy on coin tosses is indeedy informative. If you always do the complete opposite of what they say, you now do better than luck. It's like a few of my poker buddies. Some of them lose a heck of a lot more often than they win. And boy, do I love playing poker with them. Heh.

Please read my follow up. I think a medium that guesses Boy Boy Boy Girl Repeat or Heads Heads Heads Tails Repeat will be more wrong than a medium who guesses heads all the time or tails all the time (or boy all the time or girl all the time). It seems to me they're likely to be wrong more than 50% of the time and also likely to be maximally wrong.

Am I incorrect in this assessment? If so a detailed explanation would be appreciated.

 

[Dave1001]

Exactly. You were right before the edit. It the outcomes are 50/50, then the optimal solution is to pick either/or, it doesn't make a difference. So if you pick boys 50% of the time, or 70% of the time, or 3% of the time, you'll still be "accurate", in total, 50% of the time. If one probability is slightly higher than the other, then the optimal solution is to ALWAYS bet on the higher one.

(Note: Current user is perverse and should be illegal)

But it seems like I'm also picking a frequency (3:1) that is incorrect, rather than the correct frequency of (1:1). If that has no impact, I'd like a more detailed explanation as to why.

 

[RecoveringYuppy]

LOL. We seem to be second guessing ourselves to death here. I'm going with my original thought. My "second thought" was wrong. Since the predicitons are independent events it does not matter what pattern the guesses follow.

 

[Baron Samedi]

Here's the algebraic formula.

Let p=probability to guess "Boy"
therefore: 1-p = probability to guess "Girl"

If you guess "Boy", you will be correct 50% of the time.
If you guess "Girl", again, you'll be right 50% of the time.

All told, you will be right:
p*50% for your boy guesses + (1-p)*50% for your girl guesses
= 0.5p + 0.5 - 0.5p
= 0.5 = 50%

Again, this only works if the outcome is 50%. Any other outcome and it won't be this nice.

 

[Baron Samedi]

Please read my follow up. I think a medium that guesses Boy Boy Boy Girl Repeat or Heads Heads Heads Tails Repeat will be more wrong than a medium who guesses heads all the time or tails all the time (or boy all the time or girl all the time). It seems to me they're likely to be wrong more than 50% of the time and also likely to be maximally wrong.

Am I incorrect in this assessment? If so a detailed explanation would be appreciated.

I think I see why this confuses people. Let's say that we know the population is 50% B, 50% G. The only way we can possibly be completely accurate is to guess roughly 50% B 50% G. Using this strategy, you can indeed be 100% accurate, but you can also bomb out completely and be wrong every time. If you guess "Boy" all the time, you're almost guarenteed to be right 50% of the time. So the question is, do you bet safe (BBBBBBB...) or do you "go big or go home"?

 

[Baron Samedi]

LOL. We seem to be second guessing ourselves to death here. I'm going with my original thought. My "second thought" was wrong. Since the predicitons are independent events it does not matter what pattern the guesses follow.

Only on coin flips. And only if you flip coins with people who aren't cheating. Then it doesn't matter what you pick, you're screwed

 

[articulett]

Really? Can you give me an example of a single event that does not have a 50/50 chance of happening?

I understand the odds of many things happening, like flipping 6 heads in a row, but with a single event there is a 50/50 chance of that happening. Right?

Is there a 50-50 chance that any given female is pregnant? What if she's 90? What if she's 2? What if she's sterile? Even if we only counted fertile women, there is far less than a 50-50 chance that any woman tested at random will test positive for pregnancy.

You can either be pregnant or not be pregnant. There are 2 choices. However, each choice is not equally likely. 2 choices does not mean that each choice has an equal chance of being correct.

Either leprechaun's exist or they don't. Does that mean there is a 50-50 chance that they do. I think not.

 

[Baron Samedi]

Is there a 50-50 chance that any given female is pregnant? What if she's 90? What if she's 2? What if she's sterile? Even if we only counted fertile women, there is far less than a 50-50 chance that any woman tested at random will test positive for pregnancy.

You can either be pregnant or not be pregnant. There are 2 choices. However, each choice is not equally likely. 2 choices does not mean that each choice has an equal chance of being correct.

Either leprechaun's exist or they don't. Does that mean there is a 50-50 chance that they do. I think not.

Waitasecond... you're from Vegas and you're trying to help people understand probabilities in a correct way? Isn't there some law against that?

 

[The Atheist]

Thank you for this post and all the posts. I am flawed in thinking that whether or not something will happen is the same as a 50% chance of it happening. This is not correct.

I thank you all for the help. That is why I posted here. Lots of very intelligent people to help me through my flawed thinking.

Thanks to all,
Hammer

Good on ya!

This proves beyond any doubt that you're not an idiot.

If you were an ID/YEC proponent, you'd immediately be arguing the toss as to why everything is 50/50 instead of accepting the sensible answers you've been given.

Well played!

(Great to spot those "skeptics" who don't bother reading a thread before posting answers - I see they're still giving you the same old, same old, despite you already admitting you've changed your mind)

 

[RecoveringYuppy]

Dave and Baron,

Thanks for pointing out my confusion. I was definitely confused and thinking about a different problem when I started second guessing myself.

I think I see why this confuses people. Let's say that we know the population is 50% B, 50% G. The only way we can possibly be completely accurate is to guess roughly 50% B 50% G. Using this strategy, you can indeed be 100% accurate, but you can also bomb out completely and be wrong every time. If you guess "Boy" all the time, you're almost guarenteed to be right 50% of the time. So the question is, do you bet safe (BBBBBBB...) or do you "go big or go home"?

No, you can be completely accurate no matter what you do. The only question is how long can you be completely accurate. And the answer to that is that you are as likely to be correct as many times in a row no matter how you construct your guesses.

No matter how you construct your guess about the outcome of a coin toss you have a 50/50 chance of being correct. It doesn't matter if you guess the same thing every time, toss another coin, or chose any arbitrary pattern. All coin tosses are independent so you've got a 50/50 shot no matter how you decided to guess heads or tails for any individual toss. And, consequently, you will have the same kind of winning and losing streaks no matter what guessing strategy you use.

 

[Baron Samedi]

Dave and Baron,

Thanks for pointing out my confusion. I was definitely confused and thinking about a different problem when I started second guessing myself.


No, you can be completely accurate no matter what you do. The only question is how long can you be completely accurate. And the answer to that is that you are as likely to be correct as many times in a row no matter how you construct your guesses.

No matter how you construct your guess about the outcome of a coin toss you have a 50/50 chance of being correct. It doesn't matter if you guess the same thing every time, toss another coin, or chose any arbitrary pattern. All coin tosses are independent so you've got a 50/50 shot no matter how you decided to guess heads or tails for any individual toss. And, consequently, you will have the same kind of winning and losing streaks no matter what guessing strategy you use.

Okay, you got me there. There is randomness in your guessing, and randomness in the actual toss. So you're right, you CAN be completely accurate by guessing all boys. If you have 10 guesses to make, and you guess:

BBGBBGBBGB

and I guess

BBBBBBBBBB

we both have the same chances of being 100%. However, I was thinking long term and guessing over the entire population. If we know the population is 50%, then we -assume- we have to guess 50%.

I was thinking of it this way. I have 100 balls in a bag, 95 are red, 5 are blue, and we know these numbers. If we guess red all the time, we will be 95% correct. If we guess red 95 times, and blue 5 times, we can be right anywhere from 90 to 100 times.

I guess what I was trying to say with that point is that by having variance on both guessing and coin toss, you'll have more variance on the total accuracy, while always betting one way will lead to more stable results in the long run. For example, let's say that you "win" the game only if you get 13 out of 20 coin tosses, you would want more variance on the accuracy so that way you will win more often. A more stable strategy may give you results closer to the expected return more frequently. BUT I may have to take this back on the coin toss example because of this damn exact 50% outcome probability. I'll double check this. I'm just so used to never ever ever seeing a probability exactly equal to 50%, I get confused too as to the special rules (of lack thereof) with it.

 

[Ivor the Engineer]

Dave and Baron,

Thanks for pointing out my confusion. I was definitely confused and thinking about a different problem when I started second guessing myself.


No, you can be completely accurate no matter what you do. The only question is how long can you be completely accurate. And the answer to that is that you are as likely to be correct as many times in a row no matter how you construct your guesses.

No matter how you construct your guess about the outcome of a coin toss you have a 50/50 chance of being correct. It doesn't matter if you guess the same thing every time, toss another coin, or chose any arbitrary pattern. All coin tosses are independent so you've got a 50/50 shot no matter how you decided to guess heads or tails for any individual toss. And, consequently, you will have the same kind of winning and losing streaks no matter what guessing strategy you use.

Only if you assume an infinite population. I think

 

[Baron Samedi]

Only if you assume an infinite population. I think

...
I guess what I was trying to say with that point is that by having variance on both guessing and coin toss, you'll have more variance on the total accuracy, while always betting one way will lead to more stable results in the long run. For example, let's say that you "win" the game only if you get 13 out of 20 coin tosses, you would want more variance on the accuracy so that way you will win more often. A more stable strategy may give you results closer to the expected return more frequently. BUT I may have to take this back on the coin toss example because of this damn exact 50% outcome probability. I'll double check this. I'm just so used to never ever ever seeing a probability exactly equal to 50%, I get confused too as to the special rules (of lack thereof) with it.

And I stand corrected, and have to go back to my original comment. I scribbled out some numbers on the subway and worked it out properly.

- 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.

- 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.)

- If and only if N is small, and the probabilities are known for certain, and the choices are made on the entire population, and each choice is not independent, and you must be more accurate than base probability, then maybe you might want to change your strategy. For example, you have all 13 Clubs in the deck, they're dealt face down, and you must guess JQKA or not, and to win you have to get 11 right.

...
No, you can be completely accurate no matter what you do. The only question is how long can you be completely accurate. And the answer to that is that you are as likely to be correct as many times in a row no matter how you construct your guesses.

No matter how you construct your guess about the outcome of a coin toss you have a 50/50 chance of being correct. It doesn't matter if you guess the same thing every time, toss another coin, or chose any arbitrary pattern. All coin tosses are independent so you've got a 50/50 shot no matter how you decided to guess heads or tails for any individual toss. And, consequently, you will have the same kind of winning and losing streaks no matter what guessing strategy you use.

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?

 

[Myriad]

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?

I certainly do. I stop betting.

Here's the reason: 30 consecutive tosses of tails make the proposition that some kind of cheating is going on very likely.

If cheating is going on, then there's no guarantee that the results of the tosses aren't contingent on how I bet.

If I further speculate that the cheating is being done for the cheater's ultimate benefit, I would expect that as soon as I change my bets to all tails and increase the amount wagered, which might appear to be a good strategy to exploit the apparent anomaly, an excess of heads would start appearing.

As for the question about predicting boy/girl births -- again assuming that the probability of each is 50-50 -- even if you had to predict boy or girl for every birth for the entire population for a year, you're no less likely to be correct by predicting all boys as by predicting boys half the time and girls half the time. In other words, the probability of getting all your guesses right, regardless of how you guess, is just as remote as the probability of every single birth that year actually being a boy.

Respectfully,
Myriad