Topics Probability

Question Derren Brown recently had a show in which he flipped ten heads in a row. He just flipped coins all day and waited for it to happen eventually. If I flip a fair coin, I should believe there's a 50% chance it will come up heads. If I flip it three times, I should believe there's a 12.5% chance it will come up heads three times. If I have eight goes at flipping it three times, it seems I should believe there's a 100% chance of flipping three heads. If that's right, what's wrong with being increasingly confident at the beginning of each set of flips that this will be the one in which I flip three heads? It's obviously a bad argument: every time I fip the coin, there's a 50% chance it will turn up heads. But how could it be rational for me to bet that during the course of a day of coin flipping I'll flip three heads eventually but not be rational for me to be increasingly confident that the next set of three flips will be of three heads as the day progresses? Matthew

Accepted:
March 26, 2008

Comments

Yes, if I flip a fair coin 3 times I have a 1 in 23 (i.e. 1 in 8, i.e. 12.5%) chance of throwing three heads. How do we get that result? The rule is that if P and Q are independent events, then the chance of (P and Q) = chance of P x chance of Q. Likewise, if P, Q and R are independent events, then the chance of (P and Q and R) = chance of P x chance of Q x chance of R. If each of P, Q, R as a 1 in 2 chance, then the chance of (P and Q and R) is 1 in 23.

But, no, if I make 8 trials at throwing three heads I don't have a 100% chance of pulling it off. For the trials are independent events. And the chance of any one trial being successful is still 1 in 8, irrespective of what happened in the previous trials. Likewise, the chance of any one trial being unsuccessful is 7 in 8, irrespective of what happened the previous trials. So the chance of eight trials being unsuccessful is (7/8)8, which is about 0.34. So the chance of getting three heads at least once in 8 trials is .66, i.e. about two thirds (good, but very far from 100%).

Of course, the more trials at throwing three heads you make, the greater the chance of success. Thus sixteen trials will give you about a 90% chance. And the longer you go on, the nearer you get to 100%. But strictly speaking you never hit 100%: however long the sequence of trials, there remains an increasingly tiny but non-zero chance that you still fail to throw three heads every time, because the events are independent.