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Assume we have $n$ cards, indexed successively by the integers $1$ to $n$. Now each card is either marked or not. Suppose that card $i$ is marked with probability $p_i$, independently of the others.

Let the r.v. $X = 0, 1,...$ be the highest index among the marked cards (with $X = 0$ if all cards are unmarked).

What is $P(X=k)$ for $0 \leq k \leq n$ , $P(X \geq k)$ and $P(X \geq k| k \geq 1)$ for $1 \leq k \leq n$ ?

Does $X$ follow the distribution of the maximum?

For $P(X=k)$ when I compute $P(X=0)$ get $P(X=0) = \Pi_i (1-p_i)$ then $P(X=1) = \sum_{i=1}^n p_i(\Pi_{j \neq i} (1-p_j))$, but I think I am doing this wrong?

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You can think iteratively from the card numbered $n$. This one is marked with probability $p_n$. So we have the probability $$P(X=n)=p_n.$$ Now think about the case $X=n-1$ in this case, the $n^\text{th}$ card is unmarked and the $(n-1)^\text{th}$ is marked. This has probability $$P(X=n-1)=(1-p_n)p_{n-1}.$$ You probably can take it from here.

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  • $\begingroup$ Thank you. I just needed a fresh look at it. $\endgroup$ – Note Nov 17 '18 at 2:03

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