BioPHYSICAL chemistry

Given all of the probabilities for any given

event, what are the probabilities for some

outcome after a series of events? Consider

the question for the probabilities of differ-

ent outcomes after four tosses. The outcomes

can be determined by counting all possible

combinations. After tossing a coin four times,

there are a total of 16 possible outcomes

(Figure 8.1). The probability of obtaining all heads is only one in 16,

whereas the probability of obtaining at least two heads is 11 in 16. In

general, if there are Ntotal possible outcomes and the number of these

outcomes corresponding to the desired result is Ni, then the probability

of that result Piis:

(8.2)

Although the counting of all possible outcomes can be done for simple cases,

determining all possible outcomes for complex situations is not practical.

For example, consider a deck of 52 cards. How many different arrange-

ments of five cards are possible? There are 52 possibilities for the first

card, 51 for the second, 50 for the third, 49 for the fourth, and 48 for

the fifth. Each arrangement of the cards is called a permutation. The total

number of permutations is given by the product of the possible outcomes

for each card:

(52)(51)(50)(49)(48) =311,875,200 (8.3)

The product of numbers that decrease sequentially is called a factorial and

identified by an exclamation point !, assuming that the numbers decrease

to 1:

n! =n(n−1)(n −2)(n−3)... (3)(2)(1) (8.4)

The total number of possible configurations of a deck of 52 cards is then

given by 52!. But what about the previous example, of the possible con-

figurations of five cards from the deck of 52? In that case the factorials

can be used if the contributions of the remaining 47 cards are removed:

(8.5)

In general, consider a set of nobjects. The number of permutations possible

for a subset of j objects, P(n,j) is given by:

P(n,j) =(n)(n−1)(n−2)...(n−j+1) (8.6)

()()()()()

!

!

52 51 50 49 48

52

47

=

P

N

i N

= i

164 PARTI THERMODYNAMICS AND KINETICS

Figure 8.1All
possible outcomes of
four spins; this is
analogous to tossing
a coin four times.