Probability Question

Originally posted by shrew
There's a 50/50 chance that it might be me!
But at least I'm polite.

I find it extremely impolite when people reply to my posts and don't take a minimal effort to understand them.

So if I'm impolite, maybe it's not without cause.

Originally posted by Dance Master MC
This isn't really anything intellectually stimulating, but rather a question that I need to know the answer to...

My question is: If you flip a coin 3 times, and all 3 times it lands on heads, does it become more likely that it'll land on tails if you flip the coin again?

I thought that the probability still remains the same (50% chance of head and on tails when the coin is flipped again.

Can anyone offer some clarification to this?

….I thought that the probability still remains the same (50% chance of heads or tails) but my friend doesn't think so,
..…

I don’t know why but this kind of error seems to be a astonishingly common mental error amongst a large proportion of the human population -for some reason they cannot grasp the concept of probabilities not necessarily being dependent of each other (depending on the context).

I once had a friend that told me that he had completely lost a large number of times in a row with his “lucky number” on the lottery so he “reasoned” that in order for the probabilities to “even out” it would be more likely that his “lucky number” would work for him next time thus he should put loads of extra money on it! -I tied to explain to him why it doesn’t work like that but he was unconvinced and he just lost loads of his money as a result.

"If the result is HHH.....T what is the probabilty the coin is fair?"

What price would you give me that the coin is "fair" before any of the tests comence and would you be prepared to lay that price after 3 tests?

Originally posted by shrew
"If the result is HHH.....T what is the probabilty the coin is fair?"

What price would you give me that the coin is "fair" before any of the tests comence and would you be prepared to lay that price after 3 tests?

If I could put a bet down at 2:1 payout every time the result came back HHH, I would put down about 1/9 of my bankroll every time it happened, no question.

Considering that the odds are only 1:9 that the tests come back HHH and the die is fair, it's a pretty safe bet.

Originally posted by forkedknight
If I could put a bet down at 2:1 payout every time the result came back HHH, I would put down about 1/9 of my bankroll every time it happened, no question.

Considering that the odds are only 1:9 that the tests come back HHH and the die is fair, it's a pretty safe bet.

I ran a simulation of 1000 trials:
-starting with $1000
-betting 1/9 of your bankroll each time the coin flips come HHH
-running until 100 bets are made

An average of 180 trials to make 100 bets
minimum final bankroll of $225,016.20
maximum final bankroll of $15,613,635.48
average final bankroll of $3,061,788.45

Originally posted by forkedknight
I ran a simulation of 1000 trials:
-starting with $1000
-betting 1/9 of your bankroll each time the coin flips come HHH
-running until 100 bets are made

An average of 180 trials to make 100 bets
minimum final bankroll of $225,016.20
maximum final bankroll of $15,613,635.48
average final bankroll of $3,061,788.45

I noticed my simulation wasn't terribly repeatable with only 1000 trials. Here are the numbers for 10000 trials:
average of 177 trials to make 100 bets
average bankroll of $3,963,000
max bankroll of $24,020,000
min bankroll of $141,000

*edit* and I see I'm using the word 'trials' to mean two different things. There are 10,000 "trials" of starting with $1000. There was an average of 177 "trials" of flipping the coin 3 times.

Oh wow, been a while since this one was argued. In reading through it seems the debate is over the "odds" as determined by statistical analysis of a situation where one of two coins is selected and you have some toss results.

Assuming all is above board (because if it isn't we have no real means to judge an answer), then it works out this way.

Initially, we are shown there are two coins, one fair and one double-headed. One is chosen randomly using a fair process, and its identity not revealed. At this point, we only have setup information, and the odds are 50/50 for either coin.

Now the way statistical analysis works when trying to find the odds a certain assertion is either true or false is this. We assume for each case that any possibility consistent with that case could have happened. When we are told what DID happen, then we eliminate those possibilities that DIDN'T happen.

So if the coin was to come up tails, then we eliminate the 100% of the possibilities the coin is double headed, and only 50% of the time the fair coin comes up heads. What we are left looking at, statistically speaking is the 25% of the time that the fair coin would be picked and come up tails, and since that constitutes 100% of the possible remaining odds, the chance it is true is 100%.

So now, we toss the coin and it comes up heads. This happens 100% of the time with a double headed coin, but only half the time on a fair coin. We've now eliminated 25% of the possibilities, and are looking at the remaining 75%, of which 50% is the H-H coin chosen, and 25% the H-T.

Another toss and another head, and the chances of the fair coin having been chosen halves again.

Keep in mind, we don't know what was picked, we're only setting odds based on the behavior exhibited and an assumption of randomness of the experiment.

Think of it this way. The more often the coin comes up heads without coming up tails, the less likely it is that the coin and experiment are fair. You can never be totally sure, of course, but after 50 flips coming up heads, you can get pretty close to certain.

Of course, if there is ever a tails, then you are certain that the coin has a tails side.

A related question and answer? If we assume that the friend was initally right that the probablility of tails was greater than heads then we must have a tails biased coin. Against that bias, three heads have been thrown in a row. The friend is likely to have suggested there was a 75% chance of a tails on the fourth throw. If right, what are the odds of the previous three heads having been thrown? 1/64?

Originally posted by geepamoogle
Oh wow, been a while since this one was argued. In reading through it seems the debate is over the "odds" as determined by statistical analysis of a situation where one of two coins is selected and you have some toss results.

Assuming all is above board (because if it isn't we have no real means to judge an answer), then it works out this way.

Ini Of course, if there is ever a tails, then you are certain that the coin has a tails side.

this is one of the better "layman" explanations of bayesian probability that i've read. especially the use of a concrete and obvious example of how we use counter-examples to eliminate possible cases to preface a parallel explanation of why new information changes the probability of which coin we're dealing with. kudos.