Conditional Expectation on the first outcome

Question

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?

Answer 1

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.

What type of distribution is the count of Bernoulli trials until the next success?

What type of distribution is the count of Bernoulli trials until the fourth success?

What is the expectation of this something?