Question

Write p ( A ) + p ( not A ) =

Answer 1

Answer:

DEF: P(A | B) ≡ the (conditional) Probability of A given B occurs

NOT'N: | ≡ "given"

EX: The probability that event A occurs may change if we know event B has occurred.

For example, if A ≡ it will snow today, and if B ≡ it is 90° outside, then knowing that

B has occurred will make the probability of A almost zero. The probability of snow is

higher if we do not know what the temperature is. Thus, P(A | B) < P(A).

DEF: P(A | B) = P(A) ≡ A is independent of B ≡ the probability of A is unaffected by the

occurrence of event B

EX: Consider two flips of a fair coin. H ≡ Heads, and T ≡ Tails.

P(H 2nd flip | H 1st flip) = 1/2 = P(H 2nd flip). That is, knowing the outcome of the

first flip doesn't change the probability of the 2nd flip. So the two flips are

independent.

NOTE: Conditional probabilities allow us to improve our estimates of probabilities by

knowing more about the situation we are in. In elections, for example, knowing how

many people are members of each party helps us to improve the accuracy of

predictions about who will win the election. In a court case, knowing more about the

circumstances in which a crime was committed helps us judge the probability of

innocence or guilt.

Conditional probabilities allow us to reduce our sample space to just outcomes in the

event we are conditioning on. For P(A | B), we are finding the probability of A when

the sample space is restricted to B. In a Venn diagram of probabilities, we would

look only inside the area of B, and we would expand the area of B (and everything in

it) to be unity. Our total probabilities of events in B would be unity. P(A | B) would

now correspond to the size of A in B, i.e., A∩B scaled up by the same factor that

makes the size of B unity.

TOOL: The following formulas define the mathematical behavior of conditional probabilities: