Independence, Independent events, Mutually exclusive events.
Events A and B are called independent if it holds that
In the previous die-rolling example event A was rolling an even number, and event B was rolling a number greater than 2. Let's see if these are independent.
This seems to check out, so events A and B are independent.
There is yet another event, C.
Let's see if B and C are independent.
Events A and B are called mutually exclusive if it holds that
Let's see what it looks like for the events in our example.
Well, it seems these are not mutually exclusive.
On the other hand, A and C are mutually exclusive.
At an insurance company, 70% of the customers have car insurance, 60% have home insurance, and 90% have at least one of these two.
Let event A be that a customer has car insurance, and event B that a customer has home insurance. Are these two events independent?
Two events are independent if
Well, let's see what would be.
Based on these, they are not independent.
And they are not mutually exclusive either, because
At another insurance company, 80% of the customers have car insurance, and 20% of the customers have home insurance without car insurance. What percent of customers have home insurance, if having car insurance and home insurance are independent events?
Well, there is such a thing as:
So, 2/3, or 66% of customers have home insurance.
This is splendid, but let's continue with something really interesting.
We have a die that we roll once. Let event A be rolling an odd number, and event B rolling a number greater than 3.
We get the probability of A the usual way.
We count how many times it will occur and divide that by the total number of events.
There is nothing exciting so far.