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Two Beagles Probability Puzzle

"Two beagles" is a remarkable probability puzzle by Marilyn von Savant, a woman with the highest IQ in the world (218 points) who is written down in the Guinness Book of Records. She often contributes with similar probability riddles to the Parade magazine.1

Essence of the Puzzle

The essence of the puzzle is following. An owner of a pet shop informs her customer of the good news that she is about to have two beagle puppies for sale soon. The customer says OK, but he demands a dog only, not a bitch.

The pet shop owner calls an owner of the beagles who is bathing them and asks him about the gender of the puppies, respectively she asks him, whether at least one of the puppies is a dog? The owner of the puppies says yes.

The question for you is: “What is the probability of the other puppy being a dog too?”

Correct Answer

If your answer is 50:50—it means that there is an equal probability for the other puppy of being a dog or a bitch—then you guess as a majority of population. However it is not a good call. The correct answer is 1:3. How come?

There are two beagles and the number of all possibilities of combining their gender is four:

Dog Dog;
Dog Bitch;
Bitch Dog;
Bitch Bitch.

The last possibility (Bitch Bitch) can be excluded as we already know that (at least) one of the puppies is a dog. There are three possibilities remaining that fit to the fact that (at least) one of the puppies is a dog, but there is only one possibility that the other puppy is a dog too (Dog Dog).

Therefore the probability of the other puppy being a dog too is 1:3! As according to the classical definition the probability of an event is the ratio of the number of cases favorable to it (1), to the number of all cases possible (3). And that is the correct solution of this remarkable probability puzzle.

Source: 1. The puzzle and the information of its author are published in the magazine Epocha 24/2014. The analysis of the correct solution is own.

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Based on the original Czech article: Dva bíglové – pravděpodobnostní hádanka.