Part (a) The probability that a collector captures at least two small monsters of the same species is approximately 0.9941 or 99.41%.
Part (b) The lowest number of captures needed to make capturing at least two of the same species more likely than not is 2.
Part (a)
The probability of capturing at least two small monsters of the same species can be calculated by subtracting the probability of capturing five distinct species from 1 (the probability of capturing at least one of each species).
Probability of capturing five distinct species:
There are 31 choices for the first species.
After capturing one species, there are 30 choices remaining for the second species.
This continues until the fifth species is chosen, with 27 remaining choices.
Therefore, the probability of capturing five distinct species is:
(31 * 30 * 29 * 28 * 27) / (31^5)
Probability of capturing at least two of the same species:
1 - (31 * 30 * 29 * 28 * 27) / (31^5) ≈ 0.99
Therefore, the probability that a collector captures at least two small monsters of the same species is approximately 0.9941 or 99.41%.
Part (b)
To find the lowest number of captures needed to make capturing at least two of the same species more likely than not, we need to find the smallest value of k (number of captures) for which:
P(at least two of the same species) > P(exactly one of each species)
We can calculate these probabilities using the following formulas:
Probability of capturing at least two of the same species:
P(at least two of the same species) = 1 - P(exactly one of each species)
Probability of capturing exactly one of each species:
P(exactly one of each species) = (31 * 30 * 29 * ... * (31 - k + 1)) / (31^k)
We can calculate these probabilities for different values of k and compare them.
k P(at least two of the same species) P(exactly one of each species) More likely?
1 0 1 No
2 0.968 0.032 Yes
3 0.999 0.001 Yes
Therefore, the lowest number of captures needed to make capturing at least two of the same species more likely than not is 2.