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Probability

Classical Definition of Probability

Definition 2.12. The probability of an event A is defined by the formula
m
P(A) = (2.2)
n

where n is the total number of outcomes uniquely possible, equiprobable and
incompatible, m is the number of outcomes leading to the occurrence of the
event A. ✓
Example 2.1. In throwing a single unbiased die, there n = 6 mutually exclusive,
equiprobable and unique outcomes, namely getting a number of spots equal to each
of the numbers 1 through 6.

Solution. Let A be the event consisting of getting an even number of spots. Then there are
1
m = 3 outcomes leading to the occurrence of A, and hence P(A) = 3 = .
6 2

Example 2.2. From a lot of n items, k are defective. Find the probability that
l items out of a random sample of size m selected for inspection are defective.,

m
Solution. The number of possible ways to choose m items out of n is C . The favourable cases
n
are those in which l defective items among the k defective items are l selected (this can be done
l
in C ways) and the remaining m − l are nondefective, i.e., they are chosen from the total
k
l
number n − k (in C m−l ways). Thus number of favourable cases is C · C m−l . The required
n−k k n−k
probability will be
l
C · C m−l
k
P(A) = n−k .
C m
n
Statistical Definition of Probability

Let n be the total number of experiments in the whole series of trials and m be a number of
experiments in which A occurs. Then the ratio W(A) = m is called the relative frequency of
n
the event (in the given series of trials).
It turns out that the relative frequencies m observed in different series of trials are virtually the
n
same for large n, clustering about some constant
m
P(A) ∼ , (2.3)
n
called the probability of the event A.
More exactly, (2.3) means that
m
P(A) = lim .
n→∞ n
Roughly speaking, the probability P(A) of the event A equals the fraction of experiments leading
to the occurrence of A in a large series of trials.

Example 2.3. (De Mere’s paradox). As a result of extensive observation of dice
games, the French gambler de Mere noticed that the total number of spots showing
on three dice thrown simultaneously turns out to be 11 (the event A) more often
than it turns out to be 12 (the event B), although from his point of view both

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