AXIOMS OF PROBABILITY
If an experiment can result in any one of the N different equally likely
outcome, and if exactly n of these outcomes correspond to event A, then the probability of event A is
Consider an experiment whose sample space is S. For each event E
of the sample space S, we assume that a number P(E)
is defined and satisfies the following three axioms:
Axiom 1
0 ≤P(E)≤ 1
Axiom 2
P(S) = 1
Axiom 3
For any sequence of mutually exclusive events 
. . .
(that is, events for whichπΈΰ― πΈΰ―=∅ π€βππ π ≠π),
We refer to P(E) as the probability of the event E.
Thus, Axiom 1 states that the probability that the outcome of the experiment
is an outcome in E is some number between 0 and 1. Axiom 2 states that, with probability
1, the outcome will be a point in the sample space S. Axiom 3 states that, for any sequence
of mutually exclusive events, the probability of at least
one of these events occurring is just the sum of their respective probabilities.
If we consider a sequence of events 
, . . .,
where 
= S and 
= ∅ for i > 1, then, because
the events are mutually exclusive and because
we have, from Axiom 3,
implying that
π(∅)=0
That is, the null event has probability 0 of occurring.
Note that it follows that, for any finite sequence of mutually exclusive
events 
If our experiment consists of tossing a coin and if we assume
that a head is as likely to appear as a tail, then we would have
On the other hand, if the coin were biased and we felt that a head
were twice as likely to appear as a tail, then we would have
Suppose that a coin is tossed three times.
If we observe the sequence of heads and tails, then there are eight
possible outcomes S = {HHH,HHT,HTH,THH,TTH,THT,HTT,TTT}:
If we assume that the outcomes of S are equiprobable, then the probability
of each of the eight elementary events is
Let A be the event of obtaining two heads in three tosses. P(A) = P[{HHT,HTH,THH}] =
If a die is rolled and we suppose that all six sides are equally likely to appear, then we would have
P({1}) = P({2}) = P({3}) = P({4}) = P({5}) = P({6}) = 
From Axiom 3, it would thus follow that the probability of rolling an even number would equal
P({2, 4, 6}) = P({2}) + P({4}) + P({6}) = 
A batch of 6 items contains 4 defective items. Suppose 3 items are selected at random and tested.
What is the probability that exactly 2 of the items tested are defective?
A group of three undergraduate and five graduate students are available to _fill certain student
government posts. If four students are to be randomly selected from this group,
find the probability that exactly two undergraduates will be among the four chosen.
The probabilities that a secretary will make 0, 1, 2, 3, 4, or 5 or more mistake in typing a recent
report are, respectively, 0.12, 0.25, 0.36, 0.14, 0.09, 0.04 Let A = the secretary is making at most 2 mistakes.
Let B = the secretary is making at least 4 mistakes. Find P(A ∪ B) Note that, since E and 
are always
mutually exclusive and since 
we have, by Axioms 2 and 3, 1 = P(S) = P(E ∪ 
) = P(E) + P( )
*P(
) =1 - P(E)
*If πΈ ⊂πΉ then P(E) ≤P(F)
*
For married couples living in a certain suburb, the probability that the husband will vote in a school
board elections is 0.21, the probability that the wife will vote in the election is 0.28, and the probability
that they will both vote is 0.15. What is the probability that at least one will vote.
Disease I and II are prevalent among people in a certain population.
It is assumed that 10% of the population will contract disease I sometime during
their lifetime, 15% will contract disease II eventually, and 3%
will contract both diseases. Find the probability that a randomly chosen
person from this population will contract at least one disease.
From past experience a stockbroker believes that under present economic conditions
a customer will invest in tax-free bonds with a probability of 0.6, will invest in mutual funds
with a probability of 0.3, and will invest in both tax-free bonds and mutual funds,
with a probability of 0.15. At this time, _find the probability that a customer will invest
(a) in either tax-free bonds or mutual funds;
(b) in neither tax-free bonds nor mutual funds.
Consider the events
B: customer invests in tax free bonds
M: customer invests in mutual funds