Say I wanted to find what the probability of flipping a coin 1000 times and having the coin land on heads all 1000 times. How would one go about doing this?

The probability of two independent events both occurring is the product of the individual probabilities.

A single coin flip has a 1/2 chance of landing heads.
Two coin flips both landing on heads ade 1/2 * 1/2 = 1/4;

1000 flips gives (1/2)^1000.

without reading the above post, i will write that a coin has a 50% chance of getting a heads

chance of heads = .5
chance of 2 heads, that is, heads and heads = (.5*.5) = .5^2
heads, heads then heads = (.5*.5*.5)=.5^3
heads n times = .5^n
heads 1000 times = .5^1000

to find the "probabìlity of improbability" you find the probability of it not happening, then get that to minus 1
chance of getting at least 1 tail = 1- probability of getting all heads
= 1-(.5^1000)

The simplest answer is that probability and improbability are proportional to one another.

If you have 100% chance of hitting face down and 0% chance of hitting tails down then you have an equated improbability. If you have a 75% chance of hitting face down you have a 25% chance of hitting tails down.

so find the probability of one action and use the proportionality method to determine the improbability.

They're not proportional; they just add up to the same thing (1). Which is quite different.

P(X) + P(not X) = 1
Note that P(shit) = probability of event shit occurring

Also, it's usually easier to find the probability of X than not X, as not X usually encompasses more possibilities. For instance, it's easy to get the probability of all heads, but getting the probability of NOT all heads is a bitch because you have to deal with getting all heads except for one tails, all heads except for two tails, and so on.

In response to the original question, I like the answer in the first post best. Assuming event X does not influence event Y and vice versa (i.e. they're independent), it follows that:

P(X & Y) = P(X) * P(Y)

You can think of it like P(X) amount of the time event X will occur, and within those instances (hence the *) event Y will occur P(Y) amount of the time.

/thread

Which, in the real world, is pretty much zero.

In case anyone in wondering, my computer tells me that it is close to
...

.0000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00093326361850321888

...
give or take.

What program did you use to find that?

All the shitty java ones I tried locked up.

Python, locally. God bless simple computations.