Thread: conditioning to provide a general formula

1. conditioning to provide a general formula

Independent trials that result in a success with probability p and failure 1 - p are performed. Let $\displaystyle P_n$ denote the probability that n Bernoulli trials result in an even number of successes (0 considered even). Show that
$\displaystyle P_n = p(1-P_{n-1}) + (1-p)(P_{n-1})$

I am not sure what events to condition on to provide this formula. I am fairly stumped!

2. Originally Posted by FGT12
Independent trials that result in a success with probability p and failure 1 - p are performed. Let $\displaystyle P_n$ denote the probability that n Bernoulli trials result in an even number of successes (0 considered even). Show that
$\displaystyle P_n = p(1-P_{n-1}) + (1-p)(P_{n-1})$

I am not sure what events to condition on to provide this formula. I am fairly stumped!
This is a 1-term recurence. The probability of an even number of successes in n trials is the sum of a odd number of successes in the previous n-1 trials and a success in the n-th trial and the probability of an even number of successes in the privious n-1 trials and a failure in the n-th trial.

CB