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The concept of probability, focusing on the asymptotic trend towards 50/50 proportions in finite events and the independence of events. The concept of probability as a ratio, relative frequency, and the distinction between a priori and a posteriori probability. It also discusses the concept of independent and mutually exclusive events, and the calculation of joint probabilities using the additive and multiplicative laws.
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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.
Coin toss example:
Probability as Rationality raised to Mathematical Precision
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.)
Relative Frequency:
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
Probability: Another Analysis
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.
Read as “the probability of”….what ever is in the brackets.
empirical sampling with replacement probability as a limit of relative frequency
rational logical expected frequency
Example: One is either a man or a woman. Being one precludes the other.
Example: roll of the die (1,2,3,4,5,6)
P ( ) 1.
1 6 1 6
5 6
= 166
= P ( 4 or 5 ).
2 6 2 6
4 6
= 333
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.
are independent, is given as…
P A B ( , ) = P A P B ( ) ( )
Example:
P Blue ( ) =. 5
P Blue ( / male ) =. 4
/ = if, or given, this event has occurred.
P BlueandMale ( ) =. 2
Are gender and eye colour independent?
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
All possible pairs of the events
numerator is the number of favorable events
P Xhead arow P head
X ( sin ) = ( )
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
the probabilities of the two separate events, i.e.: 1/6 + 1/6 = 2/6 =.
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)