Question 569692: another friends ask you to explain how to tell if two events are dependent or independent ? how would you reply?
Answer by JBarnum(2146)   (Show Source):
You can put this solution on YOUR website! Dependent and Independent Events
The occurrence of some events may affect the probability of occurrence of others. For example, the complementary events A and A cannot occur simultaneously. If one took place the other is out of the game:
P(A|A) = 0
regardless of the probability P(A). (P(A|B) denotes the conditional probability of A assuming B.) We say that the event A is not independent of the event A (assuming P(A) ≠ 0 of couse.) And, in general, an event A is not independent of an event B iff P(A) ≠ P(A|B), i.e., if occurrence of B affects the probability of A. It may not be obvious right away, but the relationship "not independent" is symmetric: if A is not independent of B then also B is not independent of A. Formally,
if P(A) ≠ P(A|B) then P(B) ≠ P(B|A).
To see why that is so we invoke the defintion of conditional probability,
P(A|B) = P(A∩B) / P(B),
so that P(A) = P(A|B) implies P(A∩B) / P(B) = P(A), or
P(A∩B) = P(A) P(B),
which also may serve as the definition of independency of A from B. But the later relationship is symmetric! It implies
P(B) = P(A∩B)/P(A) = P(B|A),
which exactly means that B is independent of A. We see that two events A and B are either both dependent or independent one from the other. The symmetric definition of independency is this
(*) P(A∩B) = P(A) P(B).
Two events A and B are independent iff that condition holds. They are dependent otherwise.
It's a frequent misconception that the independency or dependency of two events relates to their having or not having an empty intersection. The case of A and its complement A supplies a clear example of two dependent events with empty intersection.
Another example was introduced in the discussion of conditional probabilities, where we considered the sets
Ω = {1, 2, 3, 4, 5, 6, 7, 8},
A = {1, 3, 5, 7},
B = {7, 8},
A+ = {1, 2, 3, 5, 7} and
A- = {3, 5, 7}.
We found that P(B) = 1/4 whilst
P(B|A+) = 1/5,
P(B|A) = 1/4,
P(B|A-) = 1/3.
What this says is that B and A are independent whereas B is dependent on both A+ and A-. (Note that B∩A+ = B∩A = B∩A- = {7}.)
Returning to the question of survival and life expectancy, AN is the event of a new-born reaching the age of N years. By the meaning of it, for N > M, AN⊂AM and therefore AN∩AM = AN. In particular this means that
P(AN∩AM) = P(AN)
so that P(AN∩AM) = P(AN)·P(AM) is equivalent to P(AN) = P(AN)·P(AM) which is only possible if P(AN) = 0 or P(AM) = 1. Since the latter is unrealistic and the former is unlikely for a reasonable age N, the two events AN and AM can't be independent. In fact, the probability of reaching a certain age grows with aging. As the time goes by, the size of the surviving population goes down while the number of person in a certain (advanced) age group is fixed. Thus the probability (which is the ratio of the two quantities) of getting into that group is indeed increasing with age.
in reference to: http://www.cut-the-knot.org/Probability/IndependentEvents.shtml
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