Im trying to figure out, what the probability for choosing a malicious entry-guard, depended on the amount of Relays and the fraction of malicious Relays, is.

In the tor-faq it says for using NO Entry-Guards:

Suppose the attacker controls, or can observe, C relays. Suppose there are N relays total. If you select new entry and exit relays each time you use the network, the attacker will be able to correlate all traffic you send with probability (c/n)².

So w/o Entry-Guards its pretty easy: A OP has the probability for choosing a malicious Entry-Relay of (c/n).

And for using Entry-Guards it says:

(...) Thus, the user has some chance (on the order of (n-c)/n) of avoiding profiling, whereas she had none before.

Note that this probability for using Entry-Guards is for NOT choosing an malicious Entry-Relay as a Guard. But this cited probability is not making that much scenes to me. Can somebody explain to me?

Or, to give it a general touch again: Whats the probability for choosing a malicious Relay with the use of Entry-Guards depended on the amount of Relays (n) and the fraction of malicious Relays (c) ?

Disclaimer: I know there is a lot of literature about that topic. Especially the paper "Changing of the Guards: A Framework for Understanding and Improving Entry Guard Selection in Tor" is good for understanding up- and downsides of guard-nodes (This Tor-Blog article is a good place to start, too! And for the start of all this you need to review the paper 'Locating Hidden Servers'). But all the statistics in the literature I found about it, were differing between variable guard-parameters and what I am looking for is the difference between using and not using guards. I know that there won't be a easy answer, because this differs between just named guard-parameters. But I guess and hope that we can abstract from them.

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