Here is the situation: I have a two-sided, fair coin that I'm going to flip twice in a row. Before starting the experiment, the probability of any possible result of 2 flips (head-head, head-tail, tail-head, tail-tail) is 25% because there is a 50% chance of any given result per flip, twice in a row or 1/2 * 1/2 = 1/4 = 25%

My question is what happens if I evaluate the probability of the second flip, after the first flip has already occurred. In my math class, the teacher answered that the probability of a result on the second flip is still 25%, but this doesn't make sense to me. If the coin is memoryless, why would it not be 50% that say, after having gotten a heads on the first flip, the probability of tails coming up on the second flip would be 50% since past results can't influence future outcomes?

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    $\begingroup$ Your argument is right, there may have been miscommunication. The probability of a head on the second flip is $0.5$, both before and after the first flip. $\endgroup$ – André Nicolas Dec 19 '13 at 23:34

Yes, your argument is right (your teacher may have misunderstood you). Flip 1 ($F_1$) and Flip 2 ($F_2$) are independent. The outcomes of one do not affect the other. We can also look at it in terms of conditional probability:

$p(F_2 = H \mid F_1 = T) = p(F_2 = H)$, because the outcome of the first flip does not impact the outcome of the second flip. Note that this is different from $p(F_2 = H \land F_1 = T) = p(F_2 =H) \cdot p(F_1 = T)$, which is the multiplicative rule for independent events.

(If you are not familiar, $\land$ means and.)


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