It's math class again. Suppose I toss a coin twice without allowing you to see the results. After the two flips I tell you that at least one landed tails (I swear to Kali I'm not lying). What's the probability the other toss landed tails as well? How should you bet?
I imagine that for not a few (as it would've been in my case) alarms have gone off in their heads with the PA system blaring, Achtung! Gambler's Fallacy! Gambler's Fallacy! The probability is still 1/2 since each coin flip is independent of other flips, right?
Well, sorry to say but you sounded a false alarm. The probability figure is incorrect even as indeed coin tosses are independent events. So what's going on here?
Note that what we're given is the result for one toss. We're not told whether this is for the first or second toss. That makes a lot of difference.
There are exactly four possible outcomes for two coin tosses: TT, HT, TH, HH (the first letter of each pair denotes the result of the first coin toss). Given the problem above, since at least one of the tosses landed tails, HH (getting heads on both flips) is not relevant to our case. Thus, the only possibilities left are TT, HT, and TH.
As you can see, out of the three possible outcomes there is only one where the other toss would've been tails. Therefore, it is more likely that the other toss resulted in heads. The probability it landed tails is only 1/3. So betting heads is the sensible thing to do.
If, on the other hand, we had been told that the first toss had landed tails, then the only possibilities are TH and TT. The probability of getting tails on the second toss given that the first toss landed tails is, therefore, 50%. Likewise, had we been informed that the second toss had landed tails, then the only possible outcomes are HT and TT. The probability of the first toss landing tails is, again, 50%.
The lesson here is that it is very important to be clear about what the given scenario is and what the question is actually asking. Subtle changes can make all the difference in what the solution and answer will be.
Hugh G. Gauch, Jr. 2003. Scientific Method in Practice. Cambridge, UK: Cambridge University Press. p. 212-213.
Ian Stewart. The Interrogator's Fallacy.