Modern Engineering Thermodynamics

18.7 Introduction to Mathematical Probability 741
18.7 INTRODUCTION TO MATHEMATICAL PROBABILITY
If there are N mutually exclusive 4 equally likely outcomes of an experiment, M of which results in event A, then
we write the probability of the occurrence of event A, P A ,as
P A = M N
(18.30)
Now, if P A = 0, then event A is impossible and if P A = 1, then event A is a certainty. Generally, probabilities do
not take on these extreme values but instead lie somewhere in the region
0 ≤ P A ≤ 1
Let ~P A be the probability of event Anotoccurring. Then, because it is a certainty that either event A will occur
or it will not occur, we can write
P A + P A = 1
EXAMPLE 18.5
a. Consider the toss of a single evenly weighted die. What are the six possible mutually exclusive results that can occur?
b. Now consider the toss of a pair of evenly weighted dice and add their individual results. What is the probability of each
of the resulting sums?
c. What is the most probable sum in item b?
Solution
a. Since the die is an evenly weighted cube, all six sides have the same probability of landing face up, so we have N = 6.
Consequently the six possible mutually exclusive results are
Also, we can write
P 1 = P 2 = P 3 = P 4 = P 5 = P 6 = 1=6
P 1 = P 2 = P 3 = P 4 = P 5 = P 6 = 5=6
b. The total number of combinations of results is shown in Table 18.4. From this table, it is seen that N = 6 × 6 = 36.
Using this table, we can construct an event vs. frequency table, as shown in Table 18.5. Thus, we can compute the
following probabilities for the sum of the results of the individual die:
P 0 = P 1 = 0 P 5 = P 9 = 4=36 = 1=9
P 2 = P 12 = 1=36 P 6 = P 8 = 5=36
P 3 = P 11 = 2=36 = 1=18 P 7 = 6=36 = 1=6
P 4 = P 10 = 3=36 = 1=12
c. From Table 18.5, it is clear that in the toss of two evenly weighted dice, the number 7 is the most probable outcome,
appearing on the average of once every six tosses. It has a probability given by Eq. (18.30) of
P 7 = 1=6 = 0:1667 = 16:67%
Exercises
13. If you toss an evenly weighted die, what is the probability that it will come up with an even number? Answer: P(even
number) = 1/2.
14. Suppose you have a four-sided instead of a six-sided die. How many outcomes for the sum when tossing two such dice
are possible, and what is the most probable outcome for the sum and what is its probability? Answer: M = 16, and
most probable sum = 5 (from 1 + 4, 4 + 1, 2 + 3, and 3 + 2), then P(most probable) = P 5 = 4/16 = 0.25.
15. The probability of X not occurring is simply P(not X) = 1 − P(X). Using this concept, determine the probability of not
throwing a 7 as the sum of two six-sided dice. Answer: P(not 7) = 1 − P(7) = 1 − 1/6 = 5/6.
(Continued )
4 Mutually exclusive means that no two of the N outcomes can occur simultaneously.