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Lecture 5

GGR270H1 Lecture Notes - Lecture 5: Standard Score, Normal Distribution, Put On

Course Code
Damian Dupuy

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Lecture 5 – October 10
Probability II ***
Understanding of the chances of something happening
Probability focuses on the occurence of an event or not happening
Where one of several possible outcomes could result
Outcome are (and must be) mutually exclusive
either happens or it doesnt
CANNOT have 3 or 2 possibilities
Can be thought of as frequency of an event occuring relative to all other outcomes
P(A)= F(A)/ F(E) where....
P(A) = Probability of outcome A occuring
F(A) = Abosoute frequency of A
F(E) = Frequency of all outcomes
Probability – Example 1
Die has 6 faces numbered 1-6
What is the likelihood of rolling a 6 in one throw?
P(6) = 1/6 or 1 in 6 or 0.167 (if you multiply 0.167 by 100 = 16.7 % chance)
Same probability exists for each of the other outcomes too.
Probability – Example 2
Examine the record of wet and dry days over a 100 day period
62 days recorded dry
38 days recorded as wet
What is the Probability of a wet day occuring/
P(wet) = # of wet days/ total # of days = 38/100 = .38
Can also say 38% chance a wet day will occur.
Alternatively, Probability of a Dry day occuring is...
P(dry) = 62/100 = .62
Can also say 62% chance to get dry days
Probability 3 – RULES
Maximum probability of an outcome is 1.0
all Probabilities msut add up to 1 or ...
0.0 < P(A) < 1.0
First rule
Addition Rule
Used when finding probability of a single independent events
P (A or B) = P(A) + P (B)
anytime you have ' or' you dealing with addition rule
Probability – Addition Rule (when you see (the word – Or) know its addition rule) (PUT ON
P(a or b) = P (a) + P (b)
What is the probability of rolling a 6 or a 5 in a single throw?
P (6) = .167
P (5) = .167
Therefore Probability of a 5 OR a 6 = P (6) + P (5) or 0.167 + 0.167 = 0.334
or a 33.4 % chance of throwing a 6 or a 5 in a single throw
Probability of throwing a 5 or 6 or 4 = .167 + .167 +.167 = .501
Probability 4 - Rules continued
Used when finding probability of multiple independent events
P (A and B) = P(A) x P(B)
example what is the probability of throwing a die and getting two 6s or one 5 and one 6
Probability – Multiplication Rule
P(a and b) = P (a) x P (b)
What is the probability of rolling two sixes in subsequent throws?
P(6) = .167
P(6) = .167
Therefore, the probability of a 6 and a 6 = P(6) x P (6) or .167 x .167 = .02778
Or a 2.8% chance of throwing a 6 or a 5 in a single throw
Probability of throwing 3 sixes in a row = .167 x .167 x .167 = .00463 or approx 1/2 of 1%
Why you need to know everything mentioned above?
Probability and Probability Distributions
Probability Distributions
Often see consistent or typical patterns of probabilities in certain situations
These are called Probability Distributions
Similar to frequency distributions
(DIFFERENCE) Y axis contains probabilit of outcomes rather than frequency of
Discrete and continuous
3 key types
Desrete probability distribution
Used to determine the probability of multiple events in independent trials
There is only 2 possible OUTCOMES in anytime
Each indepent event has only 2 possible outcomes
e.g rain/no rain OR flood/ no flood
Probability of even occuring is = P
Probability of event not occuring = 1-p = q
Discrete probability distribution
Used when looking at events that occur randomly in space and time.
Especially used for distributions over space particularly with quadrat analysis of point
patterns (Point patterns - if the event occurs, you can measure on the map where tornado
touches down and then measure the probability of the places it was touched)
e.g you can map where tornados touched and assign probabilities where it could happen
Also used when probability of an event occuring is less than it not occuring
Most commonly applied distribution
It is the basis for sampling theory and statistical inference
Mathematical formula is complex
But, easier to understand via a "symmetric graph"
Provides theoretical basis for sampling and statistical inference
Need to look at the 'area under curve'
Total area under the curve represents 100% of possible outcomes
50% of values lie to the right of the mean, and 50% to the left symmetry
Normal Distribution
Bell shaped Curved Graph
Probability Distribution - Normal III
Need a methodology to effectively determine probability of values on the distribution
a way of actually getting to there
Could use integral calculus
Easier to use Table of Normal Values (Z score Table)
it gives you Z scores associated with proportions under normal curve
Observations must be standardized, to use the table
anytime you have raw data – USE THE Z SCORE