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The probability to log on to a computer from a remote terminal is $0.70$. Let $X$ denote the number of attempts that must be made to gain access to the computer. Find:

$a)$ The distribution of $X$ and prove that it is correctly defined.

$b)$ The expected value of attempts needed to gain access (+ calculation).

$c)$ The probability that at most $33$ attempts, respectively at least $22$ attempts must be made to gain access.

So the probability that I log in in the first attempt is $0.70$, the probability that I log in in the second attempt is $(1−0.7)⋅0.7=0.21$, the prob. that in the third attempt is $(1−0.7−0.21)⋅0.7=0.063$ so on...What kind of distribution is this?

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    [Geometric distribution](https://en.wikipedia.org/wiki/Geometric_distribution) the first one mentioned on the linked page. Btw, how did you get the probability on success in the third attempt? It should be $0.3^2\times0.7$. – 2017-02-01

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I think the distribution you are looking for is called the Geometric distribution. If $XX=k$ ($k=1,2,3,...$) is the number of attempts made until a successful access to the computer and $p$ is the probability to succeed in each attempt, then the probability to succeed after $k$ attempts is $Pr[XX=k]=(1-p)^{k-1}p$. You can think about it as the probability to have $k-1$ attempts where you fail followed by an attempt where you succeed.