Analisis Financiero

English
There are many occasions when we would like to determine the probabilities of events under circumstances that place a restriction on the sample space of the experiment. The effect of this restriction or condition is to reduce the sample space of the experiment to one of its event spaces. The probability of an event subject to such a condition is called a CONDITIONAL PROBABILITY.Suppose you agree to gamble on the outcome of experiment in which a coin is tossed and then a die is rolled. Each outcome is identified by an ordered pair of symbols in which the first symbol represent the outcome of the coin toss and the second represents the result of the roll of the die. The terms of the gamble provide that you win every time a head appears on the die. We let H represent the event"head on the coin" and C represent the event "a number greater than four on the die." The sample space of this experimient consists of twelve equally likely outcomes as follws:
Coin result Die Result
1 2 3 4 5 6
H (H,1) (H,2) (H,3) (H,4) (H,5) (H,6)
T (T,1) (T,2) (T,3) (T,4) (T,5) (T,6)

The event "head on the coin"contains all of the outcomes in which the first symbol in the ordered pair is H. Since six out of twelve equally likely outcomes contain H, P(H) = 6/12 = 1/2. Note that this is the probability of getting a head on the coin regardless of what occurs on the die. Similarly, the event "number greater than four on the die" contains all of the outcomes in which the second symbol in the ordered pair iseither a five or a six. Since four out of twelve equally likely outcomes correspond to this event, P(C) = 4/12 = 1/3.

* In general, for any two events A and B defined on a sample space, the conditional probability of one event given that the other occurs is obtained by dividing the joint probability of the two events by the probability of the conditional event. That is:

P(A | B) = P(A n B)P(B | A) = P(A n B)
P(B) P(A)

* Probabilities such as P(A), P(B), and P(C) are generally called marginal probabilities. A marginal probability is the probability of a single event defined on ly with reference to the sample space of the experiment. A joint probability involves two or moreevents in the same sample space, and a conditional probability is defined with reference to a restricted sample space.

P(A) = n(A) , P(B) = n(B) , P(A n B) = n(A n B)
n(s) n(s) n(s)

If we want to determine the probability of B given that A also occurs, the subset corresponding to the event of interest must bethe intersection (A n B), since both A and B must occur. The statement "that A also occurs" does not imply that A must occur before B, after B, or simultaneously with B, but simply that A must occur. However, since the occurrence of A is specified as a condition, the probability must be determined with respect to the event space of the conditional event. Therefore, the conditional probability of Bgiven A is:

n(A n B) = P(A n B) = P(B | A)
n(A) P(A)

Note that this is simply the joint probability divided by the marginal probability of the conditional event.

We frequently use Venn diagrams to illustrate the relationships among subsets or event sets.
Now consider the event B. B has also been partitioned into four mutually exclusive subsets that can beidentified as (A1 n B), (A2 n B), (A3 n B), and (A4 n B). Since the subset B must be equal to the sum of all its parts,

B = (A1 n B) + (A2 n B) + (A3 n B) + (A4 n B)

And therefore,

P(B) = P(A1 n B) + P(A2 n B) + P(A3 + n B) + P(A4 + n B)

The right-hand side of the previous equation consists of the sum of several joint probabilities.

In general, if an event B is partitioned into n...