Independent Events
Two events are independent if the occurrence of one does not affect the probability of occurrence of the other.
If two events, A and B, are independent then their joint probability is multiplicative:
An Example
A simple example of coin tosses demonstrates this. If a coin is tossed twice, there are four possible combinations of results for which side of the coin is facing up in the first and second toss:
head tail
head head
tail tail
tail head
If we let
A = first toss
B = second toss
then the probability of each of these four result combinations is:
Let's see why. For the first toss, there are only two possible results:
head
tail
If the coin toss is fair (no bias in the toss is introduced) there's a .50 probability of getting a head facing up. This is because of the finite number of possible results, in this case two - head or tail. No other outcome is possible. This is called a finite geometry, any geometric system that has only a finite number of points, in our case two. We are assuming that the coin cannot land on its edge, so it can only land in one of two ways, showing a head or tail facing up.
Let's say the coin is tossed and a head is the result:
head
This means that the first event is now in the past and cannot be changed. This is due to the arrow of time, a concept that says time only moves forward, not backward. So at this point, there is no possibility that the first toss produced a tail. There's a zero probability that the first toss is a tail. This eliminates two of the previously possible results:
tail tail
tail head
We are now left with only two possible outcomes for our experiment:
head head
head tail
The second toss result is not effected by the first toss result. The toss results are independent. Again, there's a .50 probability of a head or tail in the second toss. Let's say we get this result:
tail
Now that this result has occurred, it also cannot be changed due to the arrow of time.
So our final result is:
head tail
Looking at the tosses and possible results from a point in time before the tosses were made, we can see that our four possible results each had a .25 probability of happening:
head tail: .25 probability
head head: .25 probability
tail tail: .25 probability
tail head: .25 probability
Why? Because the second toss probabilities were only .50 of .50. The first toss had already occurred and eliminated two of the potential result combinations, leaving only two of the four available total results.
You can see these results visually with the coin combinations pictured below:
Of the four possible combinations, only one is possible with two tosses of the coin. Since each combination is equally possible, each has a .25 probability of happening.