Everyone know the chances of a proper tossed tossed coin landing on heads or tails is 50/50.
But if you and a friend were tossing a coin and it honestly, coincidentally came up, say, heads 5 times in a row and he bet you 2 to 1 that the next toss resulted in tails, for 20 bucks, would you take it? I’m not talking mathmatics, I’m talking about real life and your money.
I think “honestly, coincidentally” is the key here. If he tossted the coin thousands of times and it became evident that the coin was fair, and eventually happened to produce 5 heads in a row, then the probability the next toss will be tails is still 50%. The coin, real or mathematical, has no memory.
If, on the other hand, he came up to me and proceeded to immediately produce 5 heads, then I would be very suspicious.
Doesn’t matter how many times it showed up heads before. Chances are exactly the same.
So, yes, I would take the bet.
But I’d like to say that when you toss a coin, there is not a 50/50 chance of the results
What if the coin doesn’t land? Someone might catch it, an explosion nearby could vaporize it, etc.
What if it landed on its edge? I like to call edge when I don’t care about whatever decision is being made.
So the odds are 49/49/1 or something like that. But since you wouldn’t lose if someone caught it…
I’d still take the bet.
I fully realise the coin has no memory. I believe in mathmatical probability. But the chances of it coming up SIX times in a row ,and here’s the key phrase, at the outset wouldn’t sway you a little?
There is a always a 50/50 probability IF the coin is fair. You could figure out the Chi squared of the coin buy flipping it a whole lot of times, or just using those to see the probability of the null hypothesis being correct. Don’t mind me, we’re doing this in math class, if you’d like a full explanation just ask and ye shall recieve.
Kitty
Am I the only one a little turned on by a woman that can talk about chi squared distributions with authority? Oh, wait. Yeah, I probably am.
The fair coin is at the heart of so many probability puzzles, but in real life are coins really fair? I wonder if anyone has ever actually studied this. For the record, I take that bet, and gladly.
Rrrrrrrraaaaaoooowwww. Uh, I mean, no, you’re not the only one. Hey, Kitty, wanna do the Extended Euclidean Algorithm with me?
And, yeah, I’d take that bet. If it’s a fair coin, and it’d just shown five heads in a row, there’s a 50% chance you’ll see a sixth on the next flip. P(A|B) = P(A&B)/P(B). 0.015625/0.3125 = 0.5.
My dad once told me that most American coins are not fair, but I can’t remember if he said heads is the side that comes up more often.
But certainly, if a coin actually comes up heads 51 times out of a hundred, taking the bet described makes sense.
51 heads out of a hundred is not significant. 51,000,000 heads out of 100,000,000 would be. If you’re assuming that the coin is fair, then it’s 50-50, by definition. Of course, if a coin has come up heads a large number of times (more than just five… Ten or fifteen, maybe), it’s reasonable to conclude that it’s not fair.
I know we’re straying from the OP, but I thought I’d throw my two cents in 
I remember in statistics in college being told that some professor (maybe at CMU?) had had his students flip millions of coins, and carefully record the results. Eventually they discovered a very slight bias.
In any event, the fact that somebody is offering you a bet should affect your assessment of the probabilities involved
BTW, I just love this quote from the original post:
“I’m not talking mathmatics, I’m talking about real life and your money.”
If you accept that the coin has no memory, then you must also accept that each toss is independent of the previous tosses. The coin doesn’t know that it has just produced 5 heads, so why is it going to suddenly favor tails?
I find the most intuitive way to think about it is: the head-head-head-head-tail sequence is just as likely (or unlikely) as five heads in a row. So the probability is 50%.
<< I fully realise the coin has no memory. I believe in mathmatical probability. But the chances of it coming up SIX times in a row ,and here’s the key phrase, at the outset wouldn’t sway you a little? >>
I’m gonna answer from a slightly different persepctive than scr did. Yes, the chances of a coin coming up six heads in a row is very small… IF you are ready to make to the first toss. If you’ve already got five heads in a row, then the chances of the sixth head is still 50/50.
Think as follows: what is the chance that a newborn baby will live to be 90 years old? (answer: about 10%).
What is the chance that an 89-year old person will live to be 90 years old? (answer: about 80%, because that person has already lived through 89 of the 90 years.)
Hence, with your coin. Once you’ve made it through five heads, the chance of the sixth head is still 50%.
Isn’t that known as the “Gambler’s Fallacy” where they say “Well it can’t come up heads AGAIN”, when in reality there is only the 50/50 chance, like everyone’s been saying. The casinos love that one.
Thanks for the input guys, but I know what the ODDS are. My OP was more a psych question : Would just knowing the pure math give you confidence enough to bet?
I think for the board I’ll conduct the experiment.
I’ll start flipping a coin. Every time it comes up the same 3 times in a row I’ll keep track of the 4th flip and from that point start again. Just for giggles. I’ll get back to you.
As pointed out, the question is not one of probabilities. Assuming a non-trick coin, the odds are one out of two that you’ll get heads or tails on any flip of the coin, regardless of the coin’s flip history. The question was, would you take the bet? As the OP mentions, this is a 2:1 bet; i.e., either the proprietor of the coin walks with $20 or I walk with $40.
So the potential upside to this transaction is a 2:1 return on investment with a pretty solid 2:1 risk/reward ratio. Would I take this deal? No. Bad investment, you can find better. I look for a post risk ROI of 4:1, 3:1 will sometimes fly, but let’s stick with 4:1. That means the coin guy needs to offer me a bet wherein he pays me $160 if he loses (M(inimum)ROI*Risk), but I only pay him $20 if I lose. I’d do that deal, maybe even four times.
OK, so if said terms are offered we are compelled to look again at the machinery of the deal. We would undoubtably call for a neutral coin and flipper, suspecting that nobody in their right mind would proffer such a bet with a truly non-trick coin.
Oh, wait, this is a “friend,” right?