Help with probability

and that is why I always use the carrot even with superscript coding0.5^{^20} x 0.5 = 0.5^{^21} = 1/2097152 = approx. 0.000000477

Perhaps you can do the experiment yourself using only two coins.Set down the first coin as heads, and repeatedly flip the second coin and count the percentage of times you have matching coins. You should find matches at 50 percent of the time.If as a second set, you instead flip two coins repeatedly you will get matching head about 1/4 of the time.My guess though is that you'll understand the issue after only a few throws and that you won't even get around to doing the second set of throws.

Yes. No. I understand but don't you see the conflict. As I stand poised to flip the coin for the 20th time the chances of another head are both 50/50 and some miniscule chance of it being the 20th head. I am wondering how they can both be correct. How can the same event have more than one probability?edit I suppose that the 2 probabilities are not describing the same event even though there is only one event.Edited by ProtoTypical, : No reason given.Edited by ProtoTypical, : No reason given.

That's confused.Sure, the probability of 20 heads in a row is small. However, that is not relevant to your problem.What is relevant, is the conditional probability of 20 heads in a row, given that you already have 19 consecutive heads. And that conditional probability is 0.5 (or 50%).

The probability of getting 20 heads when you've not got any yet is minuscule. The probability of getting 20 heads when you've already got 19 is fifty-fifty.

There are twenty events, not one. Each coin flip is an event. It has a 50/50 chance of being a head. The probability that twenty consecutive events will all have a pre-specified outcome is clearly going to be different than just the probability of one event having such an outcome. Does that help?It reminds me of the Conjunction fallacy, and maybe that will help drive it home:Jane is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more likely?
Jane is a teacher
Jane is a teacher and is a feminist activist
Jane is a teacher and a rapist
Jane is a teacher and sings in a choir and wears pink shoes, and was diagnosed with Adiposis dolorosa and likes country music and dislikes redheads, is a member of secret crime fighting organisation and lost a finger to gangrene at the age of 11.(it's the first one)Edited by Modulous, : No reason given.

OK I see my mistake. The probabilities are the same when taken from the same point in the chain.So in the context of evolution. The probability of evolving a brain is pretty small but the probability of having a beneficial mutation is much bigger. The evolution of the brain is only really unlikely when viewed from the beginning of it's evolution.

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Yes. No. I understand but don't you see the conflict. As I stand poised to flip the coin for the 20th time the chances of another head are both 50/50 and some miniscule chance of it being the 20th head. I am wondering how they can both be correct. How can the same event have more than one probability?

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I think you've got it by now but no. If you already have 19 heads, getting another head automatically means that it's the twentieth. The difference is the probability of getting the 19 heads in a row in the first place.And that's how the math works - the probability would decline much more rapidly if there was some factor acting against streaks of heads or tails.Added: Just to make things clearer.The probability of the individual flips doesn't change with each flip you take. The probability of completing the sequence does. To take the obvious example if you're trying to throw 20 heads in 20 flips as soon as you throw a tail, the probability of success goes to zero. It can't be done because 1 of your 20 flips is a tail. Once you accept that it should be easier to see that as long as you keep getting heads the probability of success increases, for exactly the same reason.Edited by PaulK, : No reason given.

Yes I see that now. Thanks Paul.

Your error was somewhat related to the mistake creationist make when the calculate the probability of life forming from chemicals. That's one motivation for not letting you continue to make the error.

It is funny because that is the very point that I was trying to support on another forum and was researching the fallacy. Somehow I got fixed on the idea of 2 different probabilities for one event.

Odds of getting:HTTHT is 0.5^5TTTHH is 0.5^5HHTTT is 0.5^5The odds of getting a sequence of 5 heads and tails with the odds of 0.5^5 after flipping the coin 5 times? 1 in 1The very act of flipping the coin 5 times ensures that you will get an outcome with a probability of 1 in 32. The same for evolution. The very act of random mutation and natural selection ensure that you will get a highly improbable outcome because something has to evolve.

I like the fact that you used "Gambler's Fallacy" as the title of this post.One of the things that interests me is the psychology of the pitch, how sales works, how advertising works . . . all of that comes together in casinos.My favorite is the board at the roulette table where they list the results of the last 10 trials. I have seen tons of people betting big if the board shows 5 or 6 of the same color in a row. I just shake my head. In the long run, they get excited because they think they are cheating the system which is an example of the "perfect" sales pitch.

I once won 35,000 by tracking and sector betting in roulette. It was fake money. The dealer was a novice. We were playing for about an hour and about 90% of the spins would land in the same wheel sector as the previous spin. No idea if it was coincidence or an actual manifestation of the 'dealer's signature', but it was fun.I worked there a while, my favourite question was about 'card counters' in blackjack. The Casino's attitude was '99% of people that card count, do it badly enough but with enough confidence that it's profitable to us to allow the 1% to slip by unless they're taking the piss'.

I believe that roulette wheels have been shown to be statistically non-random due to small physical imperfections. Thrown in a dealer who becomes a bit robotic and it worsens. The MIT blackjack crew did pretty well, if memory serves. Took a ton of hands, but they raked it in. I can understand why casinos would be on the lookout for crews, but the little guy making a 100 bucks is probably safe.