Question

Question about the geometric distribution

case 1

let say S=0.6 ,F=0.4

the probability that I succeed in the fifth trial is 0.4^4*0.6=0.015

Case.2

let says S=0.4,F= 0.6

the probability that I succeed in the fifth trial is 0.6^4*0.4=0.065

my question is that how come even if I have higher successful probability 0.6  but the probability succeed in the fifth trial will be less than case 2 which has less successful probability 0.4

please explain, in my understanding is that I should use less trial to succeed if I have higher successful probability

Answer 1

Case 1: P(success in 5^th trial) = 0.4⁴• 0.6 = 0.01536

Case 2: P(success in 5th trial) = 0.6⁴• 0.4 = 0.05184

Although the probability of success in higher in first case, but it depends on which trial we need the success, i.e. for case 1, since S is higher which means it is highly probable that we will get the success in initial trial as compared to case 2. Another thing, mode of geometric distribution is 1 irrespective of value of S. So, larger the value of S, faster the probability will decrease and vice versa. So if S is large and you take 1st or 2nd trial, your probability will be higher or equal in some case for 2nd trial and goes on decreasing. Larger value of S will decrease faster so as we move further, the one with larger S will definitely get the very small value.