Question

Drop bears are large carnivorous Australian marsupials, that are known to attack campers by dropping onto tents during the night. On any given night, all drop bears only drop once, and successfully land on a tent with probability p, independently of other nights and other drop bears.

If a drop bear can successfully land on a tent two nights in a row, it earns the right to reproduce. However, if a drop bear fails two nights in a row before two consecutive successes, then it is condemned to not reproduce. What is the probability that a drop bear is condemned to not reproduce?

Answer 1

Probability of a drop bear successfully dropping is

Probability of a drop bear failing to drop is

The probability that a drop bear has failed to drop on two consecutive nights is

The probability that a drop bear has failed to drop on two consecutive nights before two consecutive success till night is:

The probability that a drop bear is condemned to not reproduce is: