Probability

• …a random experiment is an action or process that leads to one of several possible outcomes. For example:

Random Experiment…

Experiment Outcomes

Flip a coin Heads, Tails

Exam Marks Numbers: 0, 1, 2, ..., 100

Assembly Time t > 0 seconds

Course Grades F, D, C, B, A, A+

Probabilities…List the outcomes of a random experiment…List: “Called the Sample Space”Outcomes: “Called the Simple Events”

This list must be exhaustive, i.e. ALL possible outcomes included.Die roll {1,2,3,4,5} Die roll {1,2,3,4,5,6}

The list must be mutually exclusive, i.e. no two outcomes can occur at the same time:Die roll {odd number or even number} Die roll{ number less than 4 or even number}

Sample Space…

A list of exhaustive [don’t leave anything out] and mutually exclusive outcomes [impossible for 2 different events to occur in the same experiment] is called a sample space and is denoted by S.

The outcomes are denoted by O1, O2, …, Ok

Using notation from set theory, we can represent the sample space and its outcomes as:

S = {O1, O2, …, Ok}

Requirements of Probabilities…

• Given a sample space S = {O1, O2, …, Ok}, the probabilities assigned to the outcome must satisfy these requirements:

(1) The probability of any outcome is between 0 and 1

• i.e. 0 ≤ P(Oi) ≤ 1 for each i, and

(2) The sum of the probabilities of all the outcomes equals 1

• i.e. P(O1) + P(O2) + … + P(Ok) = 1

Approaches to Assigning Probabilities…

There are three ways to assign a probability, P(Oi), to an outcome, Oi, namely:

Classical approach: make certain assumptions (such as equally likely, independence) about situation.

Relative frequency: assigning probabilities based on experimentation or historical data.

Subjective approach: Assigning probabilities based on the assignor’s judgment. [Bayesian]

Classical Approach…

If an experiment has n possible outcomes [all equally likely to occur], this method would assign a probability of 1/n to each outcome.

Experiment: Rolling a dieSample Space: S = {1, 2, 3, 4, 5, 6}Probabilities: Each sample point has a 1/6 chance of

occurring.What about randomly selecting a student and observing their gender? S = {Male, Female}Are these probabilities ½?

Classical Approach…Experiment: Rolling 2 die [dice] and summing 2 numbers on top.

Sample Space: S = {2, 3, …, 12}

Probability Examples:

P(2) = 1/36

P(6) = 5/36

P(10) = 3/36

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

basic concepts

• probability of event = p0 <= p <= 1

0 = certain non-occurrence

1 = certain occurrence

• .5 = even odds• .1 = 1 chance out of 10

• if A and B are mutually exclusive events:P(A or B) = P(A) + P(B)

ex., die roll: P(1 or 6) = 1/6 + 1/6 = .33

• possibility set:sum of all possible outcomes

~A = anything other than A

P(A or ~A) = P(A) + P(~A) = 1

basic concepts (cont.)

• discrete vs. continuous probabilities• discrete

– finite number of outcomes

• continuous– outcomes vary along continuous scale

basic concepts (cont.)

0

.25

.5

discrete probabilities

p

HH TTHT

0

.1

.2

p

-5 50.00

0.22

continuous probabilities

0

.1

.2

p

total area under curve = 1

independent events• one event has no influence on the outcome

of another event• if events A & B are independent

then P(A&B) = P(A)*P(B)• if P(A&B) = P(A)*P(B)

then events A & B are independent• coin flipping

if P(H) = P(T) = .5 thenP(HTHTH) = P(HHHHH) =.5*.5*.5*.5*.5 = .55 = .03

• if you are flipping a coin and it has already come up heads 6 times in a row, what are the odds of an 7th head?

.5

• note that P(10H) < > P(4H,6T)– lots of ways to achieve the 2nd result (therefore

much more probable)

conditional probability• Conditional probability is used to determine how two

events are related; that is, we can determine the probability of one event given the occurrence of another related event.

• Experiment: random select one student in class.• P(randomly selected student is male) =• P(randomly selected student is male/student is on 3rd

row) =• Conditional probabilities are written as P(A | B) and read

as “the probability of A given B” and is calculated as:

Conditional Probability…

• Again, the probability of an event given that another event has occurred is called a conditional probability…

• P( A and B) = P(A)*P(B/A) = P(B)*P(A/B) both are true• Keep this in mind!

CONDITIONAL PROBABILITY

In a class of 20 students 10 study French, 9 study Maths and 3 study both

French Maths

37 6

S

The probability they study Maths given that they study French

The probability of P(MF) = P(MF) / P(F) = 3/10

4

• P(B|A) = P(A&B)/P(A)• if A and B are independent, then

P(B|A) = P(A)*P(B)/P(A)

P(B|A) = P(B) • One of the objectives of calculating conditional

probability is to determine whether two events are related.

• In particular, we would like to know whether they are independent, that is, if the probability of one event is not affected by the occurrence of the other event.

conditional probability …