# Conditional Expectation on the first outcome

Assume Independent trials, resulting in one of the outcomes 1, 2, 3, 4, 5 with respective probabilities $$p_i$$ for $$i=1,2,3,4,5$$ and $$\sum_i p_i = 1$$

Let $$Z$$ be the number of trials needed until the initial outcome has occurred exactly $$5$$ times. example: if we get $$1,3,3,4,1,1,1,2,1$$ then $$Z=9$$

1. We want $$E[Z]$$

2. Find the expected number of trials needed until both outcome $$1$$ and outcome $$2$$ have occurred?

For question 1, I condition on the first outcome $$O_i = 1,...,5$$:

$$E[Z] = E\bigg[E[Z|O_i] \bigg] = \sum_i p_i E[Z|O_i]$$

I am thinking $$E[Z|O_i]=1+$$ something. I get stuck here. Any insight?

• How are you progressing? @Note – Graham Kemp Oct 31 '18 at 2:34

Indeed, it is $$1+$$ something.   You are using the Linearity of Expectation.
$$\mathsf E[Z\mid O_i]$$, is the expected time until that first outcome ($$i$$) has its fourth subsequent occurance.