Introduction to probability

LECTURE 6
Probability
Temur Makhkamov
Indira Khadjieva
QM Module Leaders
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[e-mail]
Office hours: by appointment
Room IB 205
EXT: 546

Lecture outline
The meaning of probability and relevant concepts
The basic operations of probability
Sets, combination, and permutation
Mathematical expectation

Probability
Probability is a numerical measure of the chance (or likelihood) that a particular event will occur.
Probability values are always assigned on a scale of 0 to 1:
0 indicates that an event is very unlikely to occur
1 indicates that an even is almost certain to occur

Experiment vs Sample Space
The probability is a chance or likelihood of an event to happen
An experiment is an activity with an observable result
The trials – repetition of an experiment
The outcomes – results of each trial
A sample space is the set of all possible outcomes
A sample point is an element of the sample space
An Event is a subset of the sample space

Experiment vs Sample Space

Experiment vs Sample Space

Calculation of probability
There are 6 blue, 3 red, 2 yellow, and 1 green marbles in the box.
What is the probability of picking a red marble?

Calculation of probability
There are 6 blue, 3 red, 2 yellow, and 1 green marbles in the box.
What is the probability of picking a red marble?
Thus, the probability of picking a red marbles is: 3/12=0.25 or 25%

Calculation of probability
If there are n experimental outcomes, the sum of the probabilities for all the experimental outcomes must be equal to 1
In the marble scenario,
P(blue) + P(red) + P(yellow) + P(green) =
= 0.5 + 0.25 + 1/6 + 1/12 = 1

Types of counting rules
Multiple step
Combination
Permutation

Multiple step experiment
Experiment of a sequence of k steps
Total number of experimental outcomes is the product of number of outcomes in each step
(n1)(n2)(n3)…(nk-1)(nk)
Example: let’s toss the coin twice
Total number of outcomes = (n1)(n2) =2*2=4
Thus, sample space is

Probability Tree Diagram

Probability Tree Diagram

Probability calculation

Self exam-training task:
Find the number of all possible outcomes for rolling a die three times

Combination
Example:
In this classroom, a lecturer randomly picks two of five students (let’s say Shakhzoda, Imana, Azamat, Zakariyo and Artyom) to test their knowledge of probability. In a group of five smart students, how many combinations of two students may be selected?
(sequence of selection does not matter)
Verbal solution: a lecturer may have 10 picks
Shakhzoda with Imana
Shakhzoda with Azamat
Shakhzoda with Zakariyo
Shakhzoda with Artyom
Imana with Azamat
Imana with Zakariyo
Imana with Artyom
Azamat with Zakariyo
Azamat with Artyom
Zakariyo with Artyom

Combination
Combination formula: n objects are to be selected from a set of N objects, where the order of selection is not important.
Where, n! = n·(n-1) ·(n-2)…3·2·1
Example: 5! = 5·4·3·2·1 = 120
Solution:

Permutation
Example:
In this classroom, out of these five students, let’s assume their names are Aziza, Bekzod, Charos, Daler and Erkin, how may ways do we have in order to have one interviewer and one interviewee?
Verbal solution: a lecturer may have 20 picks

Permutation
Permutation formula: n objects are to be selected from a set of N objects, where the order of selection is important.
Solution:

What Permutation can tell us ?

Operations with events
The union of events A and B is the event containing all experimental outcomes belonging to A or B or both.
The intersection of A and B is the event containing the experimental outcomes belonging to both A and B.
The complement of an event A is an event consisting of all experimental outcomes that are not in A.

Operations with events
Toss a die and observe the number that appears on top
S = {1, 2, 3, 4, 5, 6} (sample space)
A = {2, 4, 6} (even numbers)
B = {1, 3, 5} (odd numbers)
C = {2, 3, 5} (prime numbers)
AUC = {2, 3, 4, 5, 6) - the event that an even number or a prime number is observed
B∩C = {3, 5} - the event that an odd prime number is observed
CC = {1, 4, 6} - the event that a non prime number is observed
Exercises:
Find 1) BUC; 2) A∩C; 3) (BUC)C

Relationship of events
Mutually exclusive events -
The events A and B do not have any experimental outcomes in common
Dependent events
The event A has an influence on the event B
Independent events
The event A has no influence on the event B

Addition rule for union
Question: On Quantitative Methods module for 600 CIFS students at WIUT, 480 passed the in-class test and 450 passed the final exam, 390 students passed both exams.
Due to high failure rate, the module leader decides to give a passing grade to any student who passed at least one of the two exams.
What is the probability of passing this module?
The addition rule is used to compute the probability of the union of two events:

Multiplication rule for intersection

Multiplication Rule
A bowl contains 7 red marbles and 3 black marbles. Two marbles are drawn without replacement from the bowl. What is the probability that both of the marbles are black?

Solution
Let A = the event that the first marble is black;
let B = the event that the second marble is black.
We know the following:
In the beginning, there are 10 marbles in the bowl, 3 of which are black. Therefore, P(A) = 3/10.
After the first selection, there are 9 marbles in the bowl, 2 of which are black. Therefore, P(B|A) = 2/9.
Therefore, based on the rule of multiplication:
P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (3/10) * (2/9) = 6/90 = 2/30 = 0.067 or 6.67%

Mathematical expectation
365bet.com sent you following offers for coming El-Classico game depending on your betting:
If Barcelona wins, they will triple your money
If Real Madrid wins, they will double your money
If game results in a draw, they will quadruple your money
If you would like to bet for $100, assuming the possibilities of outcomes are equally likely, what will be the expected sum of your money?
Your possible earnings:
$300 or $0 if you bet on Barcelona’s victory
$200 or $0 if you bet on Real Madrid’s victory
$400 or $0 if you bet on a draw
Hence, the expected sum of your income will be

Concluding remarks
Today, you learned:
Basic concepts within probability theory
Basic operations of calculating the sample space and number of probable events (combinations, permutations)
Operations with events
Relationship of events
Multiplication and Addition Rule
Expected Value