The Tauroto region contains 31 different species of IceCream-type small monsters. Each capture of an IceCream-type small monster has an equal chance of being any of the 31 species. If a collector captures five of these small monsters, what is the probability that they have captured at least two small monsters of the same species?

You can determine the probability of the collector has captured at least two small monsters of the same species is the complent of that all the monsters captured are of different species, i.e. none are of the same species.

The probability than none of the five captured small monsters are of the same species are:

1. Probability than the one monster is of the species One, and the others are not of the species One:

(1/32) × (31/32) × (30/32) × (29/32) × (28/32)

2. Probability that the first monster is of the species Two, and the others are not of the species Two:

(1/32) × (31/32) × (30/32) × (29/32) × (28/32)

3. Probability that the first monster is of the species Three and the others are not of the same species:

(1/32) × (31/32) × (30/32) × (29/32) × (28/32)

4. Inference

As you see you have to do this 32 times, and then add all the equal 32 probabilities, which is

32 × (1/32) × (31/32) × (30/32) × (29/32) × (28/32) = 0.72

That is the probability that none of the five small monsters are of the same species. Then, the probability that at leas two small monsters are of the same species is 1 - 0.72 = 0.28