probability fundamentals
TRANSCRIPT
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Quote of the Day
Life is a school of
probability
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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.
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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
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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
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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
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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;
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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.
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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.
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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.
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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.
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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)
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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
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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
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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)
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Mutually Exclusive Events
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.
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Mutually Exclusive Events
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Non-Mutually Exclusive Events
(Over lapping events)
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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
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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 -
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Forum for discussion
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Thank You!