Probabilities

A particular type of coin is suspected of being biased towards tails if you toss it. Assume that all coins are equally biased, if they're biased at all.

Experimenter A tests this by tossing one of these coins six times. His results are:
T,T,T,T,T,H
He decides this isn't enough evidence to conclude that the coin is biased.

Experimenter B tests a coin by tossing it until she gets a heads. The coin tosses are:
T,T,T,T,T,H
She concludes that the coin is probably biased.

Both tests were done at the same significance level (ie the strength of evidence needed to show bias), both experimenters tossed the coin the same way, and both of the experimenters' calculations were correct. So why are their conclusions different?

Originally posted by Acolyte
A particular type of coin is suspected of being biased towards tails if you toss it. Assume that all coins are equally biased, if they're biased at all.

Experimenter A tests this by tossing one of these coins six times. His results ar ...[text shortened]... alculations were correct. So why are their conclusions different?

I'm not sure how to word this scientifically, but it's because the two experimenters were performing the tests under different sets of conditions. The first one knew from the start that he would flip the coin 6 times, no matter what. So the fact that it took six flips before the first "heads" result wasn't strange to him. There is always a good chance that one side of the coin will be dominant in a small number of flips. Were he to continue flipping the coin up to, say, 100 times, he knows that the result will even itself out to roughly 50/50.

Now the second experimenter only plans to flip the coin until it comes up heads. So she is specifically looking for that result. From that perspective, the fact that the coin lands 5 tails in a row when she is waiting for heads, would lead her to suspect that the coin is biased.

Originally posted by Acolyte
A particular type of coin is suspected of being biased towards tails if you toss it. Assume that all coins are equally biased, if they're biased at all.

Experimenter A tests this by tossing one of these coins six times. His results are:
T,T,T,T,T,H
He decides this isn't enough evidence to conclude that the coin is biased.

Experimenter B tests a c ...[text shortened]... both of the experimenters' calculations were correct. So why are their conclusions different?

are you giving their results in order, or all tails then all the heads

Originally posted by fearlessleader
are you giving their results in order, or all tails then all the heads

He gave all results in order. You have to pay attention to the conditions of when each experiment would end.

Originally posted by Natural Science
I'm not sure how to word this scientifically, but it's because the two experimenters were performing the tests under different sets of conditions. The first one knew from the start that he would flip the coin 6 times, no matter what. So the fact that it took six flips before the first "heads" result wasn't strange to him. There is always a go ...[text shortened]... ils in a row when she is waiting for heads, would lead her to suspect that the coin is biased.

You're right, but could you give some numbers to back this up?

Originally posted by Acolyte
You're right, but could you give some numbers to back this up?

The fact that I opened with "I'm not sure how to word this scientifically" should have hinted at the fact that, well, I'm not sure how to word this scientifically. I haven't dealt with probabilities since high school. I know there is mathematical formula, but I don't remember it. So I decided to answer the question using logic and reasoning, rather than brute calculation.

The chance of getting one or fewer heads out of six tosses is 7/(2^6) = 10%, which is pretty high.

The chance of having to toss six times before you get a heads is
(1/2)^6 = 1.6%, which is much lower.

The probability that would have to be crossed before bias was declared is probably 5%. Therefore, the second case shows bias, while the first does not.

Only tester A was correct, tester B is dealing with a population sample far to small to make any valid conclusions. Tester B is teh ghey fux0re, your riddle is teh suck.

Pwned!

Originally posted by Acolyte
A particular type of coin is suspected of being biased towards tails if you toss it. Assume that all coins are equally biased, if they're biased at all.

Experimenter A tests this by tossing one of these coins six times. His results ar ...[text shortened]... alculations were correct. So why are their conclusions different?

So - which tester had the better test?

Assume that a biased coin only throws heads on average once in a hundred times.

Tester A will presumably declare the coin biased if he throws 6 tails (otherwise his test was useless)
He has a chance of (0.99)^6 that the biased coin will do that, ie his chance of declaring the biased coin "true" is 1-(0.99)^6, about 5.85%
He has a chance of 0.5^6 that a good coin will do this, i.e his chance of declaring a good coin biased is about 1.56%

Tester B will presumably declare the coin biased if her first head is after the fifth toss.
She has a chance of 0.01+0.99*0.01 + 0.99^2*0.01 + 0.99^3*0.01 + 0.99^4*0.01 = 4.9% of declaring a biased coin true
and a chance of 0.5^5 = 3.1% of declaring a good coin biased

Now, whose method is better depends on how many coins are biased.
If 1 in every 100 coins are biased, tester A will make 16 mistakes in 1000 coins, whereas tester b will make 31 mistakes

But if 50 in every 100 coins are biased, tester A will make 57 mistakes in 1000 coins, and tester b will make 40 mistakes.

Given that most coins in circulation are fair, I think tester A's method is better in real world scenarios. However in the "real world" I can beat them both by immediately declaring all coins to be unbiased! - that way I only make 10 mistakes in 1000 coins, and I take much less time to do my test!