Solution to Riddle of the Week

Solution to Riddle of the Week #12: Licking Frogs

Difficulty level: Hard

The answer is not that licking the one frog on the stump or licking the two frogs in the clearing will give you a 50 percent chance of survival either way, though it can be tempting to think so. You can see the original question here.

SOLUTION

You should dash into the clearing and lick the two frogs, which will give you a 2/3 chance of survival, or about 66.67 percent. Licking the one frog on the stump will only give you a 1/2 chance of survival, or 50 percent.

Let's talk about this. It is pretty clear and straightforward that licking the one frog on the stump with give you a 50 percent chance of survival. Roughly half the frogs are male, and the other half female, and so any individual frog you stumble upon has a 1/2 chance of being either male or female.

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. This is incorrect.

To correctly calculate your odds of survival when licking the two frogs, you need to use conditional probability. Conditional probability allows you to calculate the likelihood of something occurring based on the information you have acquired about each possibility. Let's consider all the possible scenarios for the two frogs in the clearing, what is called the sample space:

• Both frogs are male: MM
• Both frogs are female: FF
• The frog on the left is male, and the frog on the right is female: MF
• The frog on the left is female, and the frog on the right is male: FM

The position of the frog is relevant, and the male being on the left or right accounts for two separate possibilities. 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. You didn't see which frog croaked, so you don't know which one is male and have to consider both possibilities separately. But the croaking you heard did give you some information, which is that at least one of the frogs is male. So you can eliminate the second possibility in the sample space, FF.

This leaves you with the following possibilities:

• MM
• FM
• MF

Two out of the three of these possibilities has a female frog, and so licking both of the frogs will give you a 2/3 chance of survival. There are a number of variations of this problem, including the Monty Hall problem, the Three Prisoners problem, and the Boy or Girl paradox, but perhaps the clearest example is Bertrand's Box paradox, which goes like this:

There are 3 boxes: a box containing 2 gold coins, a box containing 2 silver coins, and a box containing 1 silver coin and 1 gold coin. You chose one of the boxes at random, without any way to know which box you chose. You reach into the box and pull out a coin, without seeing the one you left in the box. The coin you picked is gold. What is the probability that the remaining coin is also gold? Again, it is not 1/2, but 2/3. Consider the possibilities:



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. Our sample space would then look like this:

• M1M2 (M1 croaks)
• M2M1 (M2 croaks)
• FM
• MF

Such a sample space would give us a 50 percent chance of survival if you were to lick both frogs.

This, however, is an incorrect interpretation because it assumes information that we are not given. 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. 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. 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.

Although there is one last consideration to make, more of an unusual observation than anything else. 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. In addition to MM, the sample space would only include either MF or FM, because you know which frog croaked. 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.

It is a curious case where having more information actually decreases the likelihood that you will achieve the desired outcome. If this just doesn't sit right with you, then you can come back next week for another riddle!