probability fundamentals

8/11/2019 Probability Fundamentals

1/25


2/25

Quote of the Day

Life is a school of

probability


3/25

Example of Probability

Problem: A spinner has 4 equal sectors

coloured yellow, blue, green and red.

What are the chances of landing on blue after

spinning the spinner? What are the chances of

NOT landing on red?

Solution: The chances of landing on blue are

1 in 4, or one fourth. The chances of not

landing on red are 3 in 4, or three fourth.


4/25

Probability Of An Event

P(A) = The Number Of Ways Event A Can OccurThe total number Of Possible Outcomes

Example 1:

A coin has two outcomes and

One way of events happening

P(A) = 1/2


5/25

Example 2

A single 6-sided die is rolled. What is the

probability of each outcome? What is the

probability of rolling an even number? of

rolling an odd number?

P(1..6) = 1/6

P(even number) = 3/6

P(odd number) = 3/6


6/25

Example 3

A glass jar contains 6 red, 5 green, 8 blue and 3yellow marbles. If a single marble is chosen atrandom from the jar, what is the probability of

choosing a red marble? a green marble? a bluemarble? a yellow marble?

P(red) = # of ways to choose red = 6

total # of marbles 22P(green) = # of ways to choose green = 5

total # of marbles 22


7/25

Possible or Impossible?

Impossible event A; P(A)=0;

Example: picking the Ace of swords out of a standard pack ofcards.

Certain event B; P(B)=1;

Example: A teacher chooses a student at random from a class of

girls. What is the probability that the student will be a girl?

P(X) must be between 0 and 1, both inclusive;


8/25

Sample Spaces

A sample space is the set of all possible outcomes. The sum ofall the probabilities in the sample space is 1.

Example: What would be the sample space for the rolling of a

standard die?

{1,2,3,4,5,6} - all the possible outcomes.

Example: What about flipping two coins?

{HH, HT, TH, TT} - all outcomes denoted by (H)eads or (T)ails.


9/25

Are Sample Spaces Unique?

Reconsidering the previous coin flipping example:

Instead of denoting the sample space using (H)eads and

(T)ails we could for example count the number of

heads in which case the sample space would be:{0,1,2} - For example HH would be equivalent to 2 in

this sample space

So an experiment can have multiple sample spaces all of

which are technically correct depending on the

modelling choices we make.


10/25

Are all Sample Spaces as Useful?

For the coin flipping example we have Sample Spaces

of:

{HH, HT, TH, TT} - all outcomes denoted by (H)eads or

(T)ails.

OR

{0,1,2} - For example HH would be equivalent to 2 inthis sample space.


11/25


12/25

Empirical Probability

Empirical probability is based on observation.

The empirical probability of an event is the relativefrequency of a frequency distribution based uponobservation.

It is the ratio of the number of "favourable" outcomesto the total number of trials.

Empirical probability is an estimateof a probability

P(E) = f / n

Example: A bird watcher logs the species that she sees.Out of the 100 birds that are recorded, 20 weresparrows therefore the estimated probability would be:20/100 or 0.2.


13/25

The Compliment of an Event

Definition: The complementof an event A is

the set of all outcomes in the sample space that

are not included in the outcomes of event A.

The complement of event A is represented

by (read as A bar).

Rule: Given the probability of an event, the

probability of its complement can be found by

subtracting the given probability from 1.

P() = 1 - P(A)


14/25

Example

A spinner has 4 equal sectors colored yellow,blue, green and red. What is the probability oflanding on a sector that is not green after

spinning this spinner?Sample Space: {yellow, blue, green, red}

Probability:P(not green) = 1 - P(green) = 1 - 1 = 3

4 4


15/25

The Compliment of an Event

A single card is chosen at random from a standarddeck of 52 playing cards. What is the probabilityof choosing a card that is not a king?

There are four kings in the sample space thereforethe probability of choosing one is 4/52.

Using the compliment rule:

P(Not King) = 14/52 = 48/52 = 12/13


16/25

Mutually Exclusive Events

Two events are mutually exclusive if they cannot both occurat the same time. Another word that means mutuallyexclusive is disjoint.

If two events are disjoint, then the probability of them bothoccurring at the same time is 0.

Disjoint: P(A and B) = 0

If two events are mutually exclusive, then the probability ofeither occurring is the sum of the probabilities of eachoccurring.

Specific Addition Rule

Only valid when the events are mutually exclusive.

P(A or B) = P(A) + P(B)


17/25


Example: What is the probability of throwing a 1 or 2using a fair 6-sided die?

P(X=1) = 1/6

P(X=2) = 1/6

P(X=1 OR X=2) = P(X=1) + P(X=2) = 1/6 + 1/6 = 2/6 Note that the two events are mutually exclusive as the

die cant be in two states at the same time.

Example: A single 6-sided die is rolled. What is the

probability of rolling a 5 or an odd number?The number rolled can be a 5 and odd. These eventsare not mutually exclusive since they can occur at thesame time.


18/25



19/25

Non-Mutually Exclusive Events

(Over lapping events)


20/25


21/25

Practice

Construction of unique examples:

Please make two sample spaces each

explaining one of the following concepts:

1. Mutually Exclusive Events

2. Mutually non-Exclusive Events


22/25

Reading

Set of 52 poker playing cards, must know:

Colors

Suits

Face cards etc.

http://en.wikipedia.org/wiki/Playing_card
http://en.wikipedia.org/wiki/Playing_cardhttp://en.wikipedia.org/wiki/Playing_card


23/25


24/25

Forum for discussion

For discussion and sharing material about the

course, please join the official Facebook group

of the course MM2014 :

https://www.facebook.com/groups/334857673342994/
https://www.facebook.com/groups/334857673342994/https://www.facebook.com/groups/334857673342994/https://www.facebook.com/groups/334857673342994/https://www.facebook.com/groups/334857673342994/


25/25

Thank You!

probability fundamentals

Documents