What is the probability that you encounter at least one luma during the 300 encounters while using a radar?

Question

In the game Temtem there is a very small (1 out of 7500) chance that you will encounter a "luma" version of a creature. With a special radar item, you can increase these chances by encountering many of the same creature in a row. A radar lasts for 300 encounters, and changes the chances in the following way: For the first 100 encounters, the luma rate is the same as normal (1 out of 7500), for the second 100 encounters, the rate is increased by 5 times (1 out of 1500), and for the last 100 encounters the rate is increased by 10 times (1 out of 750).

Answer 1

Assuming the encounters are independent:

1) The probability of *not* encountering a luma on the first 100 encounters with a radar is (7499/7500)^100 (each encounter has a probability of 7499/7500 of not encountering a luma and there are 100 of them)

2) For the second 100 encounters, the probability is (1499/1500)^100

3) For the third 100 encounters, the probability is (749/750)^100

Thus the probability of not encountering any luma creatures while using a radar for 300 encounters is:

p = (7499 * 1499 * 749)^100/(7500 * 1500 * 750)^100 ~ 0.81

Your probability of encountering at least one luma is therefore 1-p ~ 0.19 (19% or roughly 1 in 5)