The number of permutationsis the number of different ways specific entities
within the cat group can be arranged, with the positions being important. For exam-
ple, given five cats, how many unique ways can they be placed in three positions on
the shelf if position is important? The answers include ade, aed, dea, dae, ead, eda, abc,
acb, bca, bac, etc.—a total of 60 ways. The notation for this is 5 P3 5!/(5 3)! 5
4 3 2 1 / (2 1) 5 4 3 60 (P in this case stands for permutations).Combinationsmean the number of different ways specific entities can be grouped;
but in this case, position does not matter. For example, in the problem of the cats, how
many can be grouped into threes if position does not matter? The answers include abc,
abd, abe, acd, ace, ade; but groupings such as cba are not allowed since it is equal to
another combination: abc. The notation for this is 5 C3 5!/((53)! 3!) 5 4 3
2 1/(2 1 3 2 1) 5 2 10 (C in this case stands for combinations).With repeatables,position is important, too. But in this case, if one has five differ-
ent cats, and many clones of each, how many unique ways can they be placed in three
positions? This answer includes aaa, bbb, ccc, ddd, eee, eec, cee, etc.—a total of 125
ways. The notation for this is 5 R3 53 125 (R in this case stands for repeatables).What are some examplesin which probabilityis used?
There are thousands of examples in which probability is used, some are familiar, and
some originate from the seamier side of life. For example, everyone has played at coin
tossingat one time or another. Although there is no such thing as an idealized coin—
a circular one of zero thickness—most coin tosses use the coins available, with either
side face up (“heads” or “tails”; also phrased “heads up/down” or “tails up/down”).
Thus, one can think of a coin as a two-sided die in lieu of the six-sided cubes we are all
used to in a game of dice. If a coin is tossed with a good amount of spin, we can denote
the two possible results as H for heads and T for tails. If we repeat the tosses N number
of times, we obtain N (H) heads and N (T) tails. Thus, the fraction of N(H)/N and
N(T)/N can be thought of as the chance (probability) to get a head or tail, respectively;
254 P(H) and P(T) are the most common notations that represent the probability to get
What is a random walk?
A
lthough one might think of a random walk as one that a person takes on the
spur of the moment in a part of town he has never walked before, it means
something totally different in probability theory. A random walk is a random
process made up of a discrete sequence of steps, all of a fixed length. For exam-
ple, in physics, the collisions of molecules in a gas are considered a random walk
that are responsible for diffusion.