There are 4 cards with numbers 3,5,9,2 on each card. You draw card repeatedly one by one until you get card with number 2 on it (every time you put back pulled card),then game stops. How to calculate the average number of picks? Is that a law of large numbers or something like that?
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This is a textbook example of a geometric distribution. A geometric distribution with parameter $p$ ($p$ representing the probability each independent trial of a "success" occurring) is well known to have expected number $\frac{1}{p}$, in your case here $\frac{1}{1/4} = 4$. Read any textbook covering an introduction to probability to learn why. – JMoravitz Jun 21 '22 at 17:39
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You're new so you might profit from the following. First, pare your question to its essentials. The numbers $3,5,9,2$ are ARBITRARY and will confuse readers. Why those numbers? Instead use $1,2,3,4$ or $A,B,C,D$. Next always show YOUR work. – David G. Stork Jun 21 '22 at 17:44