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Probability
Seeing structure and order within chaotic, chance events.
Defining the boundaries between what is mere chance and what probably is not.
Asymptotic Trend :
However, we never reach a stable exact 50/50 proportion with a finite (limited) number of tosses.
In fact, as the number of tosses increases , the probability of an exact 50/50 proportion
decreases and approaches zero.
Coin toss example:
As you increase the number of tosses (of a coin), the gap
between the observed proportions and the expected
proportions (50/50) closes, but by progressively smaller
amounts.
Probability as Rationality raised to Mathematical Precision
Probability as a ratio: quotient
numerator
deno ator
min
If there is one white ball and three black balls in a bag and you reach in and draw one out, which colour would you bet that you picked?
There is one chance in four of picking the white ball. There are three chances in four of picking a black ball.
If we pick a ball out of the bag a large number of times (replacing each time), what percentage of the time would we select a black ball? (Assuming that we don’t know what is in the bag.)
Probability describes the structure that exists within a population of events.
Relative Frequency:
A Priori vs. A Posteriori probability
A probability equals the ratio of the number of possibilities favorable for the event over (divided by) the total number of possible events.
Emerging pattern from coin tossing
1. While the details may differ, the distribution will be symmetrically
arranged around the central values.
2. The most frequently occurring values are the central one.
3. The least frequently occurring values lie the farthest distance from the centre.
4. The relative frequencies of the intermediate values decreases in a
regular and symmetrical fashion as we move from the centre to the
periphery.
Probability: Another Analysis
Analytic View :
If an event can occur in A number of ways, and if it can fail to occur in B ways, then P(event) equals A divided by A+B. Example: event rolling an even number on a die. A =
P(event) =3/(3+3) = ½ = 0.
Relative Frequency View :
Subjective View : Belief in the likelihood of an event.
Read as “the probability of”….what ever is in the brackets.
B = 3 {1,2,5}
empirical sampling with replacement probability as a limit of relative frequency
rational logical expected frequency
Mutually Exclusive :
Example: One is either a man or a woman. Being one precludes the other.
Exhaustive : A set of events is exhaustive if the set includes ALL possible outcomes.
Example: roll of the die (1,2,3,4,5,6)
Probability can range from 0.0 1.0.
P ( ) 1.
1 6 1 6
5 6
= 166
= P ( 4 or 5 ).
2 6 2 6
4 6
= 333
= P ( ) 7
p ( , , , , , )1 2 3 4 5 6
6 6 6 6
0 6
= 1
=
Two events are mutually exclusive if the occurrence of one event precludes the occurrence of the other.
Joint Probabilities : The probability of the co-occurrence of two or more events, if they
are independent, is given as…
P A B ( , ) = P A P B ( ) ( )
Example:
Conditional Probabilities:
Blue Brown
Males 40 60
Females 60 40
P Blue ( ) =. 5
P Blue ( / male ) =. 4
/ = if, or given, this event has occurred.
P BlueandMale ( ) =. 2
Are gender and eye colour independent?
Eye Colour
Laws of Probability
Disjunctive: (A or B) P AorB (^^ )^ =^ P A (^ )^^ + P B (^ )
Conjunctive: (A and B) (^) P AandB ( ) = P A ( ) * P B ( ) Or P(A)P(B)
Conjunctive: Multiplicative Law
P(two head in two tosses of a coin) P E (^ )^ =.
=^ =
A B
T T
T H
H T
H H
All possible pairs of the events
numerator is the number of favorable events
P Xhead arow P head
X ( sin ) = ( )
Restriction: all events must be independent
What always works is, P(H/A and H/B) = P(H/A) * P(H/B if there was a H/A)
denominator is the total number of possible events
Disjunctive: Additive Law
Tossing a die: The probability of tossing a 1 or tossing a 3 is equal to the sum of
the probabilities of the two separate events, i.e.: 1/6 + 1/6 = 2/6 =.
Restriction: The events must be mutually exclusive.
A B
T T
T H
H T
H H
Given two tosses (A and B): P H Aor H B (^ )=.
NOT ½ + ½ = 1
P(H/A or H/B) = P(H/A) + P(H/B) – P(H/A and H/B)
= 0.5 + 0.5 - 0.
P(H/A and H/B) is the product of the probabilities of the two events. See previous page.
P(1) P(3) P(1 or 3)
