A recent logical puzzle from TED discussed the probability of one of two frogs being female, if we know that at least one of them is male:

This puzzle is discussed (arriving at different conclusions) here:

and puzzles of this type even have their own Wikipedia article:

People discussing this problem tend to fall into one of two camps, and are convinced that the other camp is completely wrong, but the correct answer depends not just on what information is given in a particular version of the puzzle, but also on the factors that control what information is given.

To summarise the TED puzzle, a man has to decide if he is more likely to find a female frog in a group of two, one of which croaks like a male, or a single frog that is not croaking. The problem is that we do not know if female frogs croak, the frequency of croaking, or if being in a group of two affects the rate of croaking, but all these things affect the answer.

The explanation in the TED video appears to assume that only male frogs croak, and that when in a group of two, only one will croak, although this is not stated explicitly. Based on this they calculate a probability of 2/3 for one of the two frogs being female, but only 1/2 for the single frog being female.

The discussion at the second video on the other hand states that:

… so which (if either) is right?

One way to find out is to carefully tabulate all the possibilities, and calculate the probabilities of each, which is the approach used in the video links, arriving at different answers.

A simpler approach is to set up a computer simulation of the problem, and see what numbers come out, which is what:

Frog Croaks.xlsb

does.

The spreadsheet setup is shown below:

The spreadsheet is set up to model any situation where it is required to find a female, either from a group of 2 (assumed to be on the left), or a single specimen on the right. Obviously it could also be applied to any other situation with a binary choice and a 50/50 frequency. For each sample the spreadsheet generates 1000 random numbers between 0 and 1, and assumes a male for numbers below 0.5, or female for above. The following options are provided:

- Cell C1 is the “cut-off score”. The spreadsheet generates a second random number. If this is below the cut-off score the sex of the sample is treated as being known, otherwise it remains unknown. In the frog context this is equivalent to the frog croaking or not croaking.
- Cell E1 controls whether the cut-off score is applied to males only (only males croak), or both male and female (both croak, but with different croaks).
- The spreadsheet includes a macro to do multiple repeats of the analysis, for a range of different cut-off scores. Enter the number of repeats in cell P2, press Alt-F8, and select the “Repeats” macro.
- Results for a single run are displayed in cells K2 and L2, showing the probability of a female in the group of 2, on the left, and from the single sample on the right.
- The same results from the repeat analyses are in columns S and T, and in the graph below.

The screen shot above shows results if only male frogs croak, with a cut off score (equivalent to the probability of any individual male frog being heard to croak) in the range 0.1 to 0.8. It can be seen that the probability of a female increases from just over 50% to over 83%, but it is the same for the group of two (with a single croaking frog) and the single un-croaking frog.

If we assume that females croak at the same rate as males (with a different distinctive croak) then the probability of a female frog reduces to 50% for both the pair of frogs and the single frog, as shown below. The rate of croaking now makes no difference to the probability of a female, which makes sense because it is now assumed that females croak at the same rate as males:

So are there any circumstances where the conclusion reached in the original video (that the pair of frogs with at least one male is more likely to have a female) is correct? This would require that only males croaked, they only croaked when in a pair, and there was a 50% chance of a male frog in a pair being heard to croak. Under those conditions the probability of a female in the pair of frogs would be 2/3 (as stated in the TED video), and the probability of the single frog being female would be 1/2.

None of these details are stated or implied in the puzzle statement however, so the correct answer is that the man should go for the single frog on the right, since it is easier to catch and lick one frog than two.