*** Official joining every $4/180man until I win one ***

[McMelchior]

[ QUOTE ]
The probabilities ought to be additive.

[/ QUOTE ]
Not correct.

Each tournament is an independent event. Just because you didn't win 179 in a row doesn't mean you're guaranteed to win #180.

As with any other grouping of independent events you have to calculate the risk of not winning one specific tournament:

NW(1) = (180 - 1)/180 = 179/180 = 0.9944444 ....

From there you can calculate the risk of not winning any of X tournaments:

NW(X) = NW(1) raised to the power X = NW(1) * ... * NW(1) (that reads: NW(1) multiplied with itself X times).

This gives us the chance of winning (at least) one out of X tournaments:

CW = 1 - NW(X)

We're looking for the number of (random outcome) 180/$4.40 to play to have exactly 50% chance of winning one:

X = log(0.5) / log(NW(1)) = 124.4

So to have 50% chance of winning (at least) one tournament in a completely random universe (which doesn't seem too far off when it comes to $4.40 buy-ins) you have to play a little more than 124 of them.