Players pay a 10% rake to house. Top 10% of entries place in tournament.
If Players risks whole stack on a 50% win chance. My math tells me I can risk my whole stack 3.18 times per tournament and expect to break even. I was wondering if anybody would confirm or correct me. This is a simple game breakdown analysis.

Player take a coin flip on average
Player needs to place in 11% of tournaments played.
Average payout per player that places is 10% of total pot

".5^3.18=11.03%"

Player then places in 11.03% of tournaments played

There's no fixed number of risks to take - hands are dealt randomly, sometimes you'll get good hands more often than average, sometimes less often. In order to increase the EV of your winnings, basically, flip every time when you think your pot equity is bigger than the odds offered, by a margin that you think is good enough.

You have to take a large number of good opportunities when you're dealt good cards in order to compensate for those times when you're dealt poor ones.

If the tourney has a high percentage of weaker players and is in the early stage, then flip whenever your pot equity is substantially bigger than the odds, otherwise wait for such a better-than-average opportunity to risk your stack because weak players are likely to present you with it. The deeper you advance in the tourney, the lower should be your standards for deciding if an opportunity is good, and eventually you'll have to start taking any opportunity for any flip that's any better than breakeven - there are relatively fewer weak players and relatively more competent ones left, as well as less time left to hunt for a better opportunity.

In Vol. 2 of 'Expert Heads Up No Limit Hold'em' (a book about HU which is however one of the best books about general poker theory that I've come across), Will Tipton calls this approach (taking only very profitable flips, i.e. those that have a massively positive EV in terms of chips, vs weaker players) 'winrate-maximizing play'. In relation to non-heads-up tourneys, I'd rather call it 'ROI-maximizing play'. Its purpose is to maximize the expected winnings from the entire tournament, not each of the hands constituting it, which makes sense because rake is charged on a per-tourney basis, not per-hand, and also because only a limited number of really profitable tourneys is available (you usually can't immediately enter a similar tourney easily if you bust out of the current one).

I can tell you that if you never risk your whole stack during a tournament. You would have a 100% chance of placing every tournament. So....