REVISIONSESSIONAL TEST

Variables Qualitative variables take on

values that are names or labels. The colour of a ball (e.g., red, green, blue)

Quantitative variables are numeric. They represent a measurable quantity. For example, population of a city

Discrete vs Continuous Variables

Quantitative variables can be further classified as discrete or continuous. If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable; otherwise, it is called a discrete variable.

Basic Concepts Consider following set of values:

12, 15, 21, 27, 20, 21 Mean Mode Median Variance Standard Deviation Range

Basic Concepts Consider following set of values:

12, 15, 21, 27, 20, 21 Stem-and-Leaf Plot Box-and-Whisker Plot Frequency Distribution Table Histogram

Trial A single performance of an

experiment is called a Trial. The result of a trial is called an

Outcome or a Sample Point. The set of all possible outcomes

of an experiment is called a Sample Space.

Subsets of a Sample Space are called Events.

Examples – Sample Space The set of integers between 1 and 50 and

divisible by 8 S = { x! x2 + 4x – 5 = 0 } Set of outcomes when a coin is tossed

until a tail or three heads appear Set of sampling items randomly until one

defective item is observed.

Definitions An event is a subset of a Sample Space. The complement of an event A with

respect to S is the subset of all elements of S that are not in A.

The intersection of two events A and B is the event containing all elements that are common to A and B.

The union of two events A and B is the event containing all the elements that belong to A or B or both.

Mutually Exclusive Events Two events A and B or

Mutually Exclusive or Disjoint, if A and B have no elements in common.

ProbabilityIf the Sample Space S of an

experiment consists of finitely many outcomes (points) that are equally likely, then the probability P(A) of an event A is

P(A) = Number of Outcomes (points) in A Number of Outcomes (points) in

S

Permutation A permutation is an arrangement of all or

part of a set of objects. Number of permutations of n objects is n! Number of permutations of n distinct

objects taken r at a time is nPr = n!

(n – r)! Number of permutations of n objects

arranged is a circle is (n-1)!

Permutations The number of distinct

permutations of n things of which n1 are of one kind, n2 of a second kind, …, nk of kth kind is

n! n1! n2! n3! … nk!

Problem How many permutations of 3

different digits are there, chosen from the ten digits, 0 to 9 inclusive?

A84 B 120

C504D 720

Problem In how many ways can a

Committee of 5 can be chosen from 10 people?

A252 B 2,002

C30,240 D 100,000

Independent Probability If two

events, A and B are independent then the Joint Probability is

P(A and B) = P (A Π B) = P(A) P(B)

For example, if two coins are flipped the chance of both being heads is 

1/2 x 1/2 = 1/4

Mutually Exclusive If either event A or event B or both events

occur on a single performance of an experiment this is called the union of the events A and B denoted as  P (A U B).

If two events are Mutually Exclusive then the probability of either occurring isP(A or B) = P (A U B) = P(A) + P(B)

For example, the chance of rolling a 1 or 2 on a six-sided die is  1/6 + 1/6 = 2/3

Conditional Probability Conditional Probability is the probability

of some event A, given the occurrence of some other event B.

Conditional probability is written as P(A І B), and is read "the probability of A, given B". It is defined by

P(A І B) = P (A Π B) P(B)

Conditional Probability

Consider the experiment of rolling a dice. Let A be the event of getting an odd number, B is the event getting at least 5. Find the Conditional Probability P(A І B).

Independent EventsTwo events, A and B,

are independent if the fact that A occurs does not affect the probability of B occurring.

P(A and B) = P(A) · P(B)

Complementation Rule

For an event A and its complement A’ in a Sample Space S, is

P(A’) = 1 – P(A)

Example - Complementation Rule

5 coins are tossed. What is the probability that:

a. At least one head turns upb. No head turns up

ProblemList following of vowel letters

taken 2 at a time:a. All Permutationsb. All Combinations without

repetitionsc. All Combinations with

repetitions

Probability Mass Function

 A probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.

Cumulative Distribution Function

x P(x≤A)

1 P(x≤1)=1/6

2 P(x≤2)=2/6

3 P(x≤3)=3/6

4 P(x≤4)=4/6

5 P(x≤5)=5/6

6 P(x≤6)=6/6

Probability Density Function

A probability density function (pdf) describes the relative likelihood for the random variable to take on a given value.

Discrete Distribution: Formulas

P(a < X ≤ b) = F(b) – F(a) = b xj a

Pj

j

1 Pj (Sum of all Probabilities)

Continuous Distribution: Formulas

P(a < X ≤ b) = F(b) – F(a) = b

a

vf )(

)(vf dv = 1 (Sum of all Probabilities)

dv

Uniform Distribution

f(x) = 1/b-a a ≤ x ≤ b 0 Otherwise

F(x) = x-a/b-a a ≤ x ≤ b 0 Otherwise

Review Question

Two dice are rolled and the sum of the face values is six? What is the probability that at least one of the dice came up a 3?a. 1/5b. 2/3c. 1/2d. 5/6e. 1.0

Review Question

Two dice are rolled and the sum of the face values is six. What is the probability that at least one of the dice came up a 3?

a. 1/5b. 2/3c. 1/2d. 5/6e. 1.0

How can you get a 6 on two dice? 1-5, 5-1, 2-4, 4-2, 3-3One of these five has a 3. 1/5