Not so trivial problem with expected value

[Armadillomon]

Hello!
This is a video game related question, but please, take it seriously.
Assume we have a lake consisting of 528 tiles and among them are 4 fixed tiles on which the player can catch a fish. There is also a fixed probability of catching a fish per one usage of fishing rod which is 35%. Essentially, using a rod is a Bernoulli trial with p=0.35 when fishing on a "fished" tile and p=0 on an empty tile. The player starts fishing on tile 1 and uses his rod k times. Then he moves to tile 2, uses his rod k times and repeats the procedure until he has fished on every tile in the lake. Then he simply starts again from tile 1 and so on... My problem is as following:

After how many uses of the fishing rod on average will the player finally catch a fish with respect to k

I've been thinking about this for a few days, but still can't find a solution.
Also, I'm not interested in the result itself, but I'd like to know how exactly I should approach this kind of a problem.

[ledzep]

Sounds like a geometric distribution problem as you are interested in the number of attempts before you catch your first fish.

http://en.wikipedia.org/wiki/Geometric_distribution

[Dason]

It's not quite geometric though.

If you just want an answer you could probably just simulate what you want a lot of times and then take an average.

[Armadillomon]

I'll run a simulation anyway, but I'm more interested in whether it is possible to simply calculate it or at least approximate it

[Dason]

Well we could probably make it easier to get an approximation if we make the assumption that no matter whether the fisher caught a fish on their ith cast at a location that they always cast k times (so they continued to cast k-i more times). This way we can just think about a single cast at each location where the fisher has a probability of 0 of catching where a fish isn't and a probability of (1 - (1-.35)^k) of catching a fish if there is a fish at that location.

This simplifies it a little but still doesn't seem 'easy' since there is a known number of fish and we just cycle back to the beginning if we don't get anything after fishing at every spot on the lake.

It's an interesting problem. Somebody else might chime in with an observation that simplifies the matter a lot but there do seem to be some complications that make this a not so easy problem.

If there was only one fish that would simplify things a little further but that doesn't seem to be an interesting case.

Some rough simulations (using 10,000 trials for each k) give me this table for the expected number of casts required where row 1 represents k=1, row 2 is k=2, ...

           [,1]
 [1,]  321.2680
 [2,]  373.5959
 [3,]  433.2436
 [4,]  504.7459
 [5,]  586.4787
 [6,]  686.3016
 [7,]  772.3956
 [8,]  870.2149
 [9,]  968.4931
[10,] 1056.3784