So let's say I can open up crates that have balls in it, but the amount of balls received from each crate has different percentage probability. As shown bellow:
10x balls = 40% chance,
20x balls = 30% chance,
50x balls = 20% chance,
100x balls = 7% chance,
200x balls = 2% chance,
1000x balls = 1% chance,
How can I figure out the average ball amount received per crate opened over an infinite amount of openings, or at least a large number opened so I can get a closely accurate answer.
What you are asking for is the expected value of the number of balls in a crate. It's essentially an average of all the possibilities weighted by their probabilities.
The expected value is calculated by multiplying the outcomes by the probability of each outcome.
For example, consider finding the expected value when you roll a normal 6-sided die.
There are 6 outcomes: 1,2,3,4,5, or 6.
The probabilities for each of those outcomes are all 1/6, so we can find the expected value in the following way:
$$\frac16*1+\frac16*2+\frac16*3+\frac16*4+\frac16*5+\frac16*6=\boxed{3.5}.$$
So this means that if we rolled a die an infinite number of times, and took the average value of the results of those rolls, the average would be 3.5.
Can you figure out how to apply this to your problem?