GGR270H1 Lecture Notes - Lecture 5: Standard Score, Normal Distribution, Put On
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
–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
–Multiplication Rule WHEN YOU SEE AND IS MULTIPLICATION (PUT ON EXAM)
–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
–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
Normal (REALLY IMPORTANT – YOU CANT DO NOTHING WITHOUT IT)
–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
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