Probability of a choice overlapping N times in a row or more when choosing K items from a selection of size M

Question

There are total of 13 players in a game, and 4 of them are assigned "evil" randomly. Some players have noted, that some players have been evil multiple times in a row (specifically, a player was evil 5 times in a row over 5 games). I am wondering about this specific probability, that is, what is the chance that we would observe this behavior, for any player (not just this particular player).

In addition to that, I want to know the general probability for this problem: M players, K assigned "evil", and over N games any 1 or more player being assigned "evil" N times in a row.

Bonus question, a curiosity: What about if there weren't 5 games, but L games in total, and a streak of N or more?

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The best thing I can solve for now is simply the probability that a specific player will be "evil" 5 times in a row:

$(\frac{4}{13})^5 \approx 0.2757%$

I tried some things with $\frac{\binom{12}{3}^5}{\binom{13}{4}^5}$, but that simply yields the result for a single specific person, and doesn't take into consideration the fact that multiple people could overlap too

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I attempted to code up a simulation which would simply check this (link to jsfiddle, freezes browser for a bit). My result seems to be around 3.55342%, which seems oddly close to $(\frac{4}{13})^5 \times 13= 3.58530%$, which seems surprising.

I also tried enumerating each possibility, however, it seems like this method wouldn't be fruitful, as the number of total possibilities is around $\binom{13}{4}^5 = 186865965446875$ (Or $\binom{M}{K}^L$), which is something my computer couldn't be able to handle. Possibly there are ways of optimizing this though...

Answer 1

The following solution applies to the original game: 13 players in all, 4 selected as "evil" each round, 5 rounds, and we want to know the probability that there is at least one player who is selected as evil in all 5 rounds.

There are $\binom{13}{4}$ possible ways to choose the $4$ evil players in each round and a sequence of $5$ independent rounds, so there are $\binom{13}{4}^5$ possible outcomes, all of which we assume are equally likely.

If there is a set of $j$ players who are selected as evil in all $5$ rounds, then the number of possible outcomes is $\binom{13-j}{4-j}^5$, for $1 \le j \le 4$. The set of $j$ players can be selected in $\binom{13}{j}$ ways.

So by Inclusion / Exclusion, the probability that at least one player is selected as evil in all the rounds is $$\sum_{j=1}^4 (-1)^{j-1} \binom{13}{j} \binom{13-j}{4-j}^5 / \binom{13}{4}^5 = 0.03564316 $$