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From the above axioms we can now prove various theorems on probability that are important in further work. Theorem 1-1 If A1 ( A2, then P(A1) Theorem 1-2 For every event A, 0 P(A) P(\) 1, (5) P(A2) and P(A2 A1) P(A2) P(A1).

i.e., a probability is between 0 and 1. Theorem 1-3 0 (6) i.e., the impossible event has probability zero.

The current-voltage relationship in an element may be instantaneous but not necessarily linear. The element is then modeled as a nonlinear resistor. An example is a lament lamp which at higher voltages draws proportionally less current. Another important electrical device modeled as a nonlinear resistor is a diode. A diode is a two-terminal device that, roughly speaking, conducts electric current in one direction (from anode to cathode, called forward-biased) much better than the opposite direction (reverse-biased). The circuit symbol for the diode and an example of its current-voltage characteristic are shown in Fig. 2-25. The arrow is from the anode to the cathode and indicates the forward direction i > 0 . A small positive voltage at the diode s terminal biases the diode in the forward direction and can produce a large current. A negative voltage biases the diode in the reverse direction and produces little current even at large voltage values. An ideal diode is a circuit model which works like a perfect switch. See Fig. 2-26. Its i; v characteristic is & v 0 when i ! 0 i 0 when v 0 The static resistance of a nonlinear resistor operating at I; V is R V=I. Its dynamic resistance is r V= I which is the inverse of the slope of the current plotted versus voltage. Static and dynamic resistances both depend on the operating point.

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Theorem 1-4 If Ar is the complement of A, then P(Ar) Theorem 1-5 If A 1 P(A) c (7) A1 < A2 < c < An, where A1, A2, . . . , An are mutually exclusive events, then P(A) In particular, if A P(A1) P(A2) c P(An) (8)

EXAMPLE 2.4. The current and voltage characteristic of a semiconductor diode in the forward direction is measured and recorded in the following table: v (V) i (mA) 0.5 2 10

S, the sample space, then P(A1) P(A < B) P(A2) P(An) 1 (9)

Theorem 1-6 If A and B are any two events, then P(A) P(B) P(A > B) (10)

0.6 0.11

1 heads _

More generally, if A1, A2, A3 are any three events, then P(A1 < A2 < A3) P(A1) P(A2) P(A3) P(A2 > A3) P(A3 > A1) (11) P(A1 > A2)

0.65 0.78

P(A1 > A2 > A3) Generalizations to n events can also be made. Theorem 1-7 For any events A and B, P(A) P(A > B) P(A > Br)

(12)

0.66 1.2

Theorem 1-8 If an event A must result in the occurrence of one of the mutually exclusive events A1, A2, . . . , An, then P(A) P(A > A1) P(A > A2) c P(A > An) (13)

0.67 1.7

If a sample space S consists of a finite number of outcomes a1, a2, c , an, then by Theorem 1-5, P(A1) P(A2) c P(An) 1 (14) where A1, A2, c , An are elementary events given by Ai {ai}. It follows that we can arbitrarily choose any nonnegative numbers for the probabilities of these simple events as long as (14) is satisfied. In particular, if we assume equal probabilities for all simple events, then 1 n , k 1, 2, c, n and if A is any event made up of h such simple events, we have P(Ak) P(A) (15)

Fig. 6-2

h (16) n This is equivalent to the classical approach to probability given on page 5. We could of course use other procedures for assigning probabilities, such as the frequency approach of page 5. Assigning probabilities provides a mathematical model, the success of which must be tested by experiment in much the same manner that theories in physics or other sciences must be tested by experiment.

0.68 2.6

A single die is tossed once. Find the probability of a 2 or 5 turning up. {1, 2, 3, 4, 5, 6}. If we assign equal probabilities to the sample points, i.e., if we assume that

0.69 3.9

P(1)

P(2)

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