One standard version of the story is here. But if you put it in a pot of nice comfortable water and then turn on the heat, the frog will complacently let himself be boiled. Since I found him persuasive on the big points, let me mention only a small one: the "frog in boiling water" myth that simply won't go away.Įveryone who has heard a political speech knows this story: You put a frog into a pot of boiling water, and it jumps right out.
Oddly enough, it was under these circumstances that I finally saw Al Gore's movie An Inconvenient Truth. It is a curious case where having more information actually decreases the likelihood that you will achieve the desired outcome.A twelve-hour flight from Shanghai to San Francisco has its drawbacks, but one of the plusses is the chance to catch up on a whole slew of movies. When the sample space only includes the two possibilities, MM and MF, then the probability that the two frogs includes a female is only 50 percent. In addition to MM, the sample space would only include either MF or FM, because you know which frog croaked. If you had in fact seen which frog it was that croaked, so you know which of the two is certainly a male, then your probability does fall to 50 percent. We know there was a croak, and therefore we know that one of the frogs is male we cannot extrapolate further than that without more information.Īlthough there is one last consideration to make, more of an unusual observation than anything else. But the problem says nothing about the frequency of croaking, so it may be that it's not uncommon for a male frog to go his whole life and only croak once. Specifically, this interpretation assumes that two male frogs are twice as likely to make a croaking sound than a pair of one male and one female. This, however, is an incorrect interpretation because it assumes information that we are not given. Such a sample space would give us a 50 percent chance of survival if you were to lick both frogs.
Our sample space would then look like this:
This introduces a very interesting question related to the frog licker: should we consider MM as two separate possibilities, frog M1 croaking, and frog M2 croaking? A well-argued blog post about this riddle contends that hearing a frog croak introduces the need to consider the frequency with which the male frogs croak, or in other words, if there were two males, one of them croaking is one possibility, and the other croaking is a separate possibility. So you can eliminate the second possibility in the sample space, FF. But the croaking you heard did give you some information, which is that at least one of the frogs is male. You didn't see which frog croaked, so you don't know which one is male and have to consider both possibilities separately. Actually, it would be more accurate to say that one specific frog being male while the other is female is a separate possibility from the inverse of that scenario, which is that first same frog is female while the other is male. The position of the frog is relevant, and the male being on the left or right accounts for two separate possibilities. The frog on the left is female, and the frog on the right is male: FM The frog on the left is male, and the frog on the right is female: MF Let's consider all the possible scenarios for the two frogs in the clearing, what is called the sample space: Conditional probability allows you to calculate the likelihood of something occurring based on the information you have acquired about each possibility. To correctly calculate your odds of survival when licking the two frogs, you need to use conditional probability. When considering the two frogs in the clearing, though, many people's first instinct is this: you know that one frog is male and will not save you you don't know which one is male there is one other frog, and you don't know what it is so therefore you have a 50 percent chance of survival if you lick both of the frogs.