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• Outcome. The result of one single trial of the random experiment.
• Sample space. The set of all possible outcomes of one single trial of the random experiment. We denote it Q. The sample space contains everything we are considering in this analysis of the experiment, so we also can call it the universe. In our diagrams we will call it U.
• Event. Any set of possible outcomes of a random experiment.
Possible events include the universe, U, and the set containing no outcomes, the empty set ф. From any two events E and F we can create other events by the following operations.
• Union of two events. The union of two events E and F is the set of outcomes in either E or F (inclusive or). Denoted E U F
• Intersection of two events. The intersection of two events E and F is the set of outcomes in both E and F simultaneously. Denoted E n F.
• Complement of an event. The complement of an event E is the set of outcomes not in E. Denoted E
We will use Venn diagrams to illustrate the relationship between events. Events are denoted as regions in the universe. The relationship between two events depends on the outcomes they have in common. If all the outcomes in one event are also in the other event, the first event is a subset of the other. This is shown in Figure 4.3.
If the events have some outcomes in common, but each has some outcomes that are not in the other, they are intersecting events. This is shown in Figure 4.4. Neither event is contained in the other.
If the two events have no outcomes in common, they are mutually exclusive events. In that case the occurrence of one of the events excludes the occurrence of the other, and vice versa. They are also referred to as disjoint events. This is shown in Figure
AXIOMS OF PROBABILITY
4.3 AXIOMS OF PROBABILITY
The probability assignment for a random experiment is an assignment of probabilities to all possible events the experiment generates. These probabilities are real numbers between 0 and 1. The higher the probability of an event is, the more likely it is to occur. A probability that equals 1 means that event is certain to occur, and a probability of 0 means the event cannot possibly occur. To be consistent, the assignment of probabilities to events must satisfy the following axioms.
1. P(A) > 0 for any event A. (Probabilities are nonnegative.)
2. P (U) = 1. (Probability of universe = 1. Some outcome occurs every time you conduct the experiment.)
3. If A and B are mutually exclusive events, then P(A U B) = P(A) + P(B). (Probability is additive over disjoint events.)
The other rules of probability can be proved from the axioms.
LOGIC, PROBABILITY, AND UNCERTAINTY
Figure 4.5 Event E and event F are mutually exclusive or disjoint events.
1. P(ф) = 0. (The empty set has zero probability.)
• U = U U ф and U П ф = ф. Therefore by axiom 3
• 1 = 1 + P(ф) . qed
2. P(A) = 1 — P(A). (The probability of a complement of an event.)
• U = A U A and A П A = ф . Therefore by axiom 3
• 1 = P (A) + P(A). qed
3. P (A U B) = P (A) + P (B) — P (A П B). (The addition rule of probability.)
• A U B = A U (A П B) and they are disjoint. Therefore by axiom 3
• P(A U B) = P(A) + P(A П B) .
• B = (A П B) U (A П B) , and they are disjoint. Therefore by axiom 3
• P(B)= P(A П B) + P(A П B). Substituting this in previous equation gives
• P(A U B) = P(A) + P(B) — P(A П B) . qed
An easy way to remember this rule is to look at the Venn diagram of the events. The probability of the part A П B has been included twice, once in P (A) and once in P(B), so it has to be subtracted out once.
4.4 JOINT PROBABILITY AND INDEPENDENT EVENTS
Figure 4.6 shows the Venn diagram for two events A and B in the universe U. The joint probability of events A and B is the probability that both events occur simultaneously,
JOINT PROBABILITY AND INDEPENDENT EVENTS
on the same repetition of the random experiment. This would be the probability of the set of outcomes that are in both event A and event B, the intersection A n B. In other words the joint probability of events A and B is P(A n B), the probability of their intersection.
If event A and event B are independent, then P(A n B) = P(A) x P(B). The joint probability is the product of the individual probabilities. If that does not hold the events are called dependent events. Note that whether or not two events A and B are independent or dependent depends on the probabilities assigned.
Distinction between independent events and mutually exclusive events.
People often get confused between independent events and mutually exclusive events. This semantic confusion arises because the word independent has several meanings. The primary meaning of something being independent of something else is that the second thing has no affect on the first. This is the meaning of the word independent we are using in the definition of independent events. The occurrence of one event does not affect the occurrence or nonoccurrence of the other events.