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.

No contradiction with the facts on the ground!

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