After many years of study, it comes down to solving logarithms in your head while you're trying to get to sleep.

How many rounds of Russian Roulette does it take before you have 50-50 or worse odds of surviving?

If we assume there is 1 bullet in a 6 chambered revolver, you have a 1/5 chance of having your wig pushed back, and 5/6 chance of taking another breath. The problem is that every time you play (and survive to play again), you're taking another trial with 5/6 odds, which gives you (5/6)^n probability of surviving after n rounds.

Problem is that when you multiply a number less than 1, you always get something smaller than what you started with. That means one day, you will bite the bullet. The question is, after how many tries?

We wish to solve the following for n:

(5/6)^n = 1/2

=> lg (5/6)^n = lg (1/2)

=> n * lg (5/6) = lg (1/2)

=> n * (lg(5) - lg(6)) = lg(1) - lg(2)

=> n * (lg(5) - lg(2) - lg(3)) = -1

Now here we reason that since lg() is a continuous, monotonically increasing function, by the intermediate value theorem:

2 < 3 < 4 < 5

=> lg(2) < lg(3) < lg(4) < lg(5)

= 1 < lg(3) < 2 < lg(5)

That gives us a tight enough bound for lg(3) ~ 1.5, considering the precision we need only to the nearest whole number, but the bound on lg(5) is pretty loose, so we'll need to do better.

We know that we can approximate an analytic function, as lg() certainly is, about a point "a" using its derivative:

lg(x) ~ lg(a) + lg'(x) * (x-a)

We also know that

lg'(x) = 1/(x * ln2)

where e ~ 2.7, so we take ln2 ~ 1.

Together this gives:

lg(5)

~ lg(4) + 1/5 * (5-4)

= 2 + 1.5 = 2.2

Going back to the original equation:

n * (lg(5) - lg(2) - lg(3)) = -1

=> n * (2.2 - 1 - 1.5) = -1

=> n = 1 / 0.3 = 3.3

Lets verify that we haven't wandered completely off-base by checking our calculation:

(5/6)^3 = 125 / 216 > 125 / 250 = 1/2

=> (5/6)^3 > 1/2

(5/6)^4 = (125 * 5) / (216 * 6) = 625 / 1296 < 1250 = 1/2

=> (5/6)^4 < 1/2

So there you have it: 3 rounds of everyone's favorite pastime gives you better than 50-50 odds, and 4 rounds give you worse than 50-50 odds. Running it through the calculator the next morning, you have a 57% chance of surviving 3 rounds, and 48% after 4.

Now, play a game of Russian Roulette and have a blast.

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