Independent events
Example: if the chance of having blond hair is 0.3 and the chance of having a cold is 0.2, the chance of meeting a blond-haired person with a cold is: 0.3×0.2=0.06
Not mutually exclusive events
Example: if the chance of having diabetes is 10% and the chance of being obese is 30%, the chance of meeting someone who is obese or has diabetes or both is (0.1+0.3)-0.1×0.3=0.37
I have question:
What is the difference between examples above? Why in one case we use the formula of Independent events but in another formula of not mutually exclusive events? I found these examples is similar.
I have only one guess: obesity is risk factor of diabetes mellitus and fat tissue has influence on glucose metabolism. So, obesity can tell us something about the occurrence of diabetes. Is it right guess?
And if it right… Why we should substract out? I think, probability to find person with obesity+diabetes is higher than if it would Independent conditions.
Best Answer
In your second example you use the formula:
$$P(D \cup O) = P(D) + P(O) - P(D \cap O)$$
and you also used:
$$P(D \cap O) = P(D) \cdot P(O)$$
The first formula is always true, but the second assumes that you are dealing with two independent events, which you can understand as follows:
It is always true that:
$$P(A \cap B) = P(A) \cdot P(B|A)$$
But if $A$ and $B$ are independent, you have that:
$$P(B|A) = P(B)$$
Plugging that in, we thus get:
$$P(A \cap B) = P(A) \cdot P(B|A) = P(A) \cdot P(B)$$
So yes, both examples assumed that the events are independent. ... though the independence of the second example is probably not true in real life ...