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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

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

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  • $\begingroup$ Thank you very much, with your explanation i understood my error $\endgroup$ – Methematic Dec 5 '19 at 9:55
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The way I solved this prpblem is that first I counted all the combinations, so 5!/(3!*1!*1!)=20, then all the combinations that result in fail. First we start when black is the firts marble drawn (Bxxxx) 4!/3!*1!=4, then GBxxx 3!/2!*1!=3, GGBxx 2!/1!*1!=1. At last we have GGGBR, so counting them we get 10/20 or 1/2 chance of failure, which gives us 1/2 chance of wining.I run a matlab code and it also came 1/2

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    $\begingroup$ But the Q says "keep on picking until you pick 3 green marbles". You are continuing to pick AFTER you have picked 3 green marbles. $\endgroup$ – almagest Dec 3 '19 at 19:42
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    $\begingroup$ oh,you are right, I didnt read that right $\endgroup$ – Heroofmetal Dec 3 '19 at 19:52

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