An experiment has probability p of success, q of failure and 1 - p - q of neither, If independent trials are repeated over and over until either success or failure is obtained, what is the probability of an ultimate success?
There are only two outcomes, success or failure. I don't know how you could have neither.
p + q = 1
Therefore, p = 1-q
To find the probability of an ultimate success, we can use the concept of geometric probability. In this scenario, a geometric distribution represents the number of independent trials needed to achieve the first success.
Let's break down the problem:
- The probability of success is denoted as p.
- The probability of failure is denoted as q.
- The probability of neither success nor failure is 1 - p - q.
Since independent trials are repeated until either success or failure is obtained, success can only occur in the first trial or any subsequent trial as long as failure is not obtained before that.
Therefore, the probability of ultimate success can be calculated as follows:
P(ultimate success) = P(success in the first trial) + P(success in second trial) + P(success in third trial) + ...
The probability of success in the first trial is simply p.
In the second trial, there are two possibilities:
1. The first trial resulted in neither success nor failure, with probability (1 - p - q). In this case, the probability of success in the second trial is p.
2. The first trial resulted in failure, with probability q. In this case, there will be no ultimate success.
Thus, the probability of success in the second trial is (1 - p - q) * p.
By following this logic, the probability of success in the nth trial is [(1 - p - q) * p]^(n-1) * p.
Now, to calculate the probability of ultimate success, we sum up the probabilities of success in each trial from 1 to infinity:
P(ultimate success) = p + (1 - p - q) * p + [(1 - p - q) * p]^(2) + [(1 - p - q) * p]^(3) + ...
This series converges to a finite value since the probability (1 - p - q) * p is less than 1. Using the sum of an infinite geometric series formula, the probability of ultimate success can be calculated as:
P(ultimate success) = p / (1 - [(1 - p - q) * p])
This formula gives the probability of an ultimate success based on the given probabilities of success, failure, and neither.