Modern Engineering Thermodynamics

744 CHAPTER 18: Introduction to Statistical Thermodynamics
Probabilities of related events that are not mutually exclusive have the following interpretation. Assume that
event B depends in some way on another event A. Then, the conditional probability of event B occurring given
thefactthateventA has already occurred is written P B/A .Thecompound probability that both events A and B
will occur is written as P AB . The relation between these two probabilities is the basic and logic probability statement,
written as
PAand ð BÞ = P AB = P A P B/A = P B P A/B (18.32)
If events A and B are totally independent, their probabilities are uncoupled and P A/B = P A and P B/A = P B : Then,
Similarly, the basic or logic probability statement can be written as
PAand ð BÞ = P AB = P A P B (18.33)
PAor ð BÞ = P A + P B − P AB (18.34)
If events A and B are mutually exclusive (i.e., they cannot occur simultaneously), then P AB = 0andP (A or B) = P A + P B .
EXAMPLE 18.6
In the draw of a single card from a full deck of playing cards, what is the probability that it will be an ace or a spade?
Solution
Equation (18.34) gives P (ace or spade) = P ace + P spade − P ace of spades . Now,
so
P ace = 4
52 = 1 13 , P spade = 13
52 = 1 4 , P ace of spades = 1
52
P ðace or spadeÞ = 4
52 + 13
52 − 1
52 = 16 = 0:308 = 30:8%
52
Exercises
16. Instead of drawing either an ace or a spade in Example 18.6, determine the probability of drawing either a 2 or a 3 of
any suit. Answer: P(2 or 3) = 0.154 = 15.4% of the time.
17. What is the probability of drawing an ace of spades in Example 18.6? Answer: P(ace and spade) = 0.0192 = 1.92% of
the time.
18. Find the probability of drawing four aces in a row out of a standard 52 card deck. Hint: Initially, there are four aces
scattered among the 52 cards, but after the first ace is drawn, only three aces remain among 51 cards, and so forth.
Answer: P(ace and ace and ace and ace) = 3.7 × 10 −6 .
Table 18.6 lists the probability of the different five-card poker hands.
Table 18.6 Probabilities of Five-Card Poker Hands
Name of Hand
Number of Card Combinations
Giving This Hand
Probability of the Hand
Royal flush 4 1.54 × 10 −6
Straight flush 36 1.38 × 10 −5
Four of a kind 624 0.00024
Full house 3,744 0.00144
Flush 5,108 0.00197
Straight 10,200 0.00392
Three of a kind 54,912 0.02113
Two pair 123,552 0.0475
One pair 1,098,240 0.42257
All other hands 1,302,540 0.50118
Total 2,598,960 1.00000