Lecture 2
Agenda
1. Sample Spaces and Events
2. Probability
Sample Spaces and Events
Whenever we perform any experiment it can result in several different outcomes. But before we perform the experiment we can’t exactly say which
outcome will occur.
Definition 1. The sample space S of any random experiment is the set of
all possible outcomes of the experiment.
Example
Experiment :: Toss a coin 3 times
Sample Space :: S = {HHH, HHT, HT H, T HH, HT T, T HT, T T H, T T T }
There are 23 = 8 possible outcomes. This is an example of a discrete or
countable sample space.
Example
Experiment :: Observe the height in ft of a randomly chosen UF student.
Sample Space :: S = [4, 7] i.e. all real numbers between 4 to 7
This is an example of a continuous or uncountable sample space.
Definition 2. An EVENT is any collection of sample points. In other words
any subset of the sample space S is called an event.
Example
Experiment :: Toss a coin 3 times
Sample Space :: S = {HHH, HHT, HT H, T HH, HT T, T HT, T T H, T T T }
A = Event that there is atleast one heads
A = {HHH, HHT, HT H, T HH, HT T, T HT, T T H}
B = event that there is atmost one heads
B = {HT T, T HT, T T H, T T T }
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Probability
Intuitively the probability of any event is a number between 0 and 1 which
shows us how likely the event is to occur in a single performance of the experiment. If the probability is 1 that means the event will surely occur, if
it’s 0 then the event won’t occur. And likewise if the probability moves up
from 0 towards 1 that means it becomes more likely to occur.
All this talk gives us the feeling of probability. But if we want to define
probability rigorously what do we do ?
Old fashioned way
If I give you a coin and tell you that the probability of Heads for this coin is
0.4, that means if I toss the coin a very large number of times then approximately 40% of the times Heads will occur.
Thus previously mathematicians used to define probability of an event like
this
Definition 3. Probability of an event A is a number p between 0 and 1,
such that if you run the experiment a very large number of times, then for p
proportion of times A will occur and for (1 − p) proportion of times Ā will
occur.
This is a very nice and good definition of probability. But after some time
mathematicians encountered some problems with it. So now we will learn
the new definition which we will use in this course.
The new definition
Let
S = Sample space of a random experiment
A = Collection of all possible events
Definition 4. A probability assisgnment P() for a random experiment is a
numerically valued function that assigns a value P (A) to every event A such
that the following axioms are satisfied
• P (A) ≥ 0 for any event A
• P (S) = 1
2
• If A1 , A2 , A3 , . . . is a sequence of mutually exclusive events i.e. Ai ∩Aj =
∅ for all pairs i 6= j then
P (∪∞
i=1 Ai )
=
∞
X
P (Ai )
i=1
Notice two things, firstly we are not defining the probability of an single event, rather
we are defining the probabilities for all the events together. Secondly we are not saying
how to actually calculate the probabilities, rather we are giving 3 axioms such that a function P () will be called a probability if it satisfies those axioms; as to how to provide such
a function without disturbing the 3 axioms is the headache of the mathematician involved
not of this definiton.
Properties
1. P (∅) = 0
Take A1 = S and A2 = A3 = . . . = ∅.We can check that the events are
mutually exclusive. Thus P (S) = P (S) + P (∅) + P (∅) + P (∅) + . . .
i.e. 1 = 1 + P (∅) + P (∅) + P (∅) + . . .
Now if P (∅) is even slightly positive then the LHS will be 1 and RHS
will be ∞.Hence P (∅) = 0.
2. If A ∩ B = ∅ then P (A ∪ B) = P (A) + P (B)
Simply take A1 = A, A2 = B and the rest of the Ai ’s to be ∅.
3. If A ⊂ B then P (A) ≤ P (B)
Defining and calculating the probability of an event by
sample point method
1. Define the experiment
2. Construct the sample space
3. Assign probabilites to each of the sample points, making sure that they
sum up to 1.
4. Express the event of interest as a collection of sample points
5. Find P (A) by summing probabilites of sample points in A.
3
Example
Random Experiment Choose a person from 4 persons, with no preference
to any person.
• What is the the sample space ?
Since there are only 4 possible outcomes the sample space is S =
{1, 2, 3, 4}.
• What is the set of all possible events? Recall that A which is the set of
all possible events is essentially the set of all possible subsets of S.Hence
A = {∅, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3},
{2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}
• How to assign a probability assignment P() which express our belief in
how the experiment is conducted and also satisfies the 3 axioms ?
Solution: Based on your belief in how the experiment is conducted assign
probability for each of the individual outcomes, i.e. for each point in the
sample space making sure that they add up to 1. For this experiment since
there is no preference to any person,
1
1
1
1
P ({1}) = , P ({2}) = , P ({3}) = , P ({4}) =
4
4
4
4
NOW DEFINE THE PROBABILITY OF ANY EVENT AS SUM OF
PROBABILITIES OF POINTS BELONGING TO THAT EVENT.
Foe example, if A is the event that person 1 or 2 is chosen,
P (A) = P ({1, 2}) = P ({1}) + P ({2}) =
1 1
+ = 0.5
4 4
This procedure will guarantee that the probability assignment satisfies those
3 axioms, for discrete sample spaces.
Homework :: From the book try 2.20,2.21,2.22,2.29,2.31. If you
can’t do 2.29 or 2.31 after lecture 2 don’t be discouraged we will
do do these more in lecture 3
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