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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.

Coin toss example:

Asymptotic Trend:

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.

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.

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.

Probability as a ratio:

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 probability equals the ratio of the number of possibilities

favorable for the event over (divided by) the total number of possible events.

A Priori vs. A Posteriori probability

Flipping a coin six times.

Number of heads tossed

A large number of people toss a coin six times:

What proportion of them will have tossed three heads and three tails?

What proportion of them will have tossed all heads or all tails?

Emerging pattern from coin tossing

- 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:

rational

logical

expected frequency

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 =

3 {2,4,6}

B =

3 {1,2,5}

Read as “the probability of”….what ever is in the brackets.

P(event) =3/(3+3) = ½ = 0.5

Relative Frequency View:

empirical

sampling with replacement

probability as a limit of relative frequency

Subjective View: Belief in the likelihood of an event.

Terminology

Event: that with which we are concerned

Independence:

two events are independent when the occurrence of one does NOT

influence the P(probability) of the other occurring.

Example:

In both cases, there are 100 observations

Independent

Not Independent

Mutually Exclusive:

Two events are mutually exclusive if the occurrence of one event precludes the occurrence of the other.

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.

Joint Probabilities:

The probability of the co-occurrence of two or more events, if they are independent, is given as…

Example:

Eye Colour

Conditional Probabilities:

/ = if, or given, this event has occurred.

Are gender and eye colour independent?

Probability Distribution of Discrete Variables

You can ask……….

What is the P of a 3?

Where do the fractions in the formula come from?

Laws of Probability

Disjunctive: (A or B)

Conjunctive: (A and B)

Or P(A)P(B)

Conjunctive: Multiplicative Law

P(two head in two tosses of a coin)

numerator is the number of favorable events

A B

denominator is the total number of possible events

T T

T H

H T

H H

All possible pairs of the events

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)

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 =.333

P(1)

P(3)

P(1 or 3)

Restriction: The events must be mutually exclusive.

A B

Given two tosses (A and B):

T T

T H

H T

H H

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 - 0.25

P(H/A and H/B) is the product of the probabilities of the two events. See previous page.

More than two events.

Given P(H) = .5 and P(T) = .5)

P(H/A) or P(H/B) or P(H/c)

Convert to P(not H/A)….

and, use multiplicative law,

and, subtract the product from 1.

1 - P(T/A) and (T/B) and (T/C) =

1 - .5(.5)(.5) = .875

Thus:

P(H/A) or P(H/B) or P(H/C) = .875

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