Average number of successful picks when drawing marbles

Suppose you have a bag with 5 marbles (1 red, 3 green, 1 black). You pick one marble at time without replacement.

If the marble is red we call this successful pick.

If the marble is green you can keep on picking from the bag until you pick 3 green marbles (let's say you have 3 lives and 1 green picked => lose 1 life)

If you pick the black one, you have to stop picking marbles.

How can I know the average number of successful picks? I can calculate the average total picks.

I hope someone can point me in the right direction. Feel free to ask if the question is not clear enough.

Thank you

• Possible hint. It might be easier to count the unsuccessful picks. (Suppose there is no red marble at all.) – Ethan Bolker Dec 3 '19 at 17:17
• Maybe ignore the green marbles and then consider the only case where they actually make a difference: when you pick 3 green marbles. – Jazzowner Dec 3 '19 at 17:21

For success you must pick R, GR or GGR. $$p(R)=1/5$$. $$p(GR)=(3/5)(1/4)$$ and $$p(GGR)=(3/5)(2/4)(1/3)$$. So (adding) the total prob of success in a single trial is 9/20. So if you run a large number of trials you expect success about 9 times out of 20.