Question

Astronomers estimate that as many as 100 billion stars in the Milky Way galaxy may be encircled by planets. Let p denote the probability a randomly selected solar system contains intelligent life. Assume the occurrence of life in one solar system is independent of the occurrence of life in another solar system (so no interstellar migration, for example). Do not include our solar system in the calculations. I invent a spaceship capable of interstellar travel and visit solar systems until I encounter one that contains intelligent life. If the 1,000,000th solar system I visit is the first that contains intelligent life, what is the maximum value that p can be?

Answer 1

Solution: Here concept of geometric distribution is used.

Solution | you visit 1000000th solar system is first solar system that contains the intelligent life. It means you have fail to obtain intelligent life 999999 timers before finding the first solar system that contains the intelligent life and - Geometric distribution i- Suppose we have a series of independent trials or repititions and in each trial the probability of success, & remains the same. Then the probability that there are a failures preceding the first success is given. by P. (1-6)* The scenario in vore given question matches with the Characteristic of Geometric distribution. i. Probability that you will get fail & gggggg times before visiting the first solar system that contains. intelligent life is given by PIG= P(1-P) 3333 ; where p in the probability that solar system contains intelligent life. Now, we want to find maximum value of p. for maximum value of p. apla) = 0 and all to ato pa ap where, a is the value obtained from the 'solving a pla) = 0 ap

99999 P(x) = Þ(1-)3959 999999 999998 OP) ? = 0 = (1-b) - 999999” (1-5) + 7 (1-0)999999– 999999 € (1-1399998 (1–13999999 (1-5199998 = 999999 h I n 1- b 1 Þ = = 999999 $ P(1+999999) = 166 → TTTTTTTTTTTTTTTT 1000000 l p - 106 10 PI) a þ2 - also play ao at pa 100 -: Maximum value of p is 100 ". Maxi