Session 4: Probability

Definitions

Random Experiment:

• A process leading to an uncertain outcome

Sample space:

• All possible outcomes of a random experiment

• Flip a coin, the sample space consists of 2 outcomes S = {H, T}.

Event

• One or more outcomes of an experiment.

• Any subset of outcomes in the sample space.

Simple Event (elementary event)

• is a single outcome.

Flip a coin,

• The sample space consists of 2 simple events: S = {H, T}

• An event of getting one Head: E = {H}

Flip a coin twice,

• The sample space consists of 4 simple events: S = {HH, HT, TH, TT}

• An event of getting one Head is a compound event: E = {HT, TH}

Probability

Probability

• is a numerical value ranging from 0 to 1.

• The probability of event A, denoted P(A)

• Assigning Probability:

  • Classical: equally likely outcomes

  • Empirical: experimentation or historical data

  • Subjective: judgment

Classical methods

• Used when the number of possible outcomes of the event of interest is known

Empirical methods

• relative frequency (% or how often) of actual observations of an event in experiment or historical data

• Conducting an experiment to observe the frequency with which an event occurs

Subjective Probability

• Used when classical or empirical probabilities are not available.

• circumstances change rapidly => inappropriate to assign probabilities based solely on historical data.

• Use individuals’ opinion, experience, belief to estimate the probabilities.

Laws of Large Numbers

• as the sample size increases, the empirical probabilities of a process will converge towards the classical probabilities

Probability Rules

• compute the probability of an event without knowledge of all the sample point probabilities: Complement of an Event Union of two Events Intersection of Two Events Mutually Exclusive Events

Complement of an event

• The complement of an event A is denoted by Aʹ (gren) (or Ā (ba)) and consists of everything in the sample space except event A.

• A ba là những giá trị khác A

• A and Aʹ together comprise the entire sample space:

  • P(A) + P(Ā) = 1

  • P(A) = 1 - P(Ā )

Intersection of Two Events

• probability of the intersection of two events is known as a joint probability

  

The Union of Two Events

• The union of two events A and B (denoted A ∪ B or “A or B”) represents the number of instances where either Event A or B occur or both events occur together.

  

The Addition Rule

Conditional Probability

• is the probability of event A occurring, given the condition that event B has occurred.

• Denoted P(A|B). The vertical line “|” is read as “given.”

  

Independent and Dependent Events

• Events A and B are independent when the probability/ occurrence of one event is not affected by the other event. If A and B are independent, then

• If P(A|B) ≠ ( ) , then events A and B are not independent

  

Multiplication Rule

Contingency tables with probabilities

Decision Tree

• are used to display marginal and joint probabilities from a contingency table

• 

Mutually Exclusive Events

• are those events that cannot occur at the same time.

• mutually exclusive events are dependent

• Independent events can occur at the same time, ME cannot

• you cannot conclude that because events are not independent, they will be mutually exclusive.

Bayes’ Theorem

• from a sample, special report, or a product test we obtain some additional information

• Given this information, we calculate revised or posterior probabilities

• Bayes’ theorem provides the means for revising the prior probabilities

  

General Forms of Bayes’ Theorem

• Suppose two events have prior probabilities P(A1) and P(A2) . ►A1 and A2 are mutually exclusive events.

• Assume the conditional probabilities P(B|A1) and P(B|A2) are known, then

• 

Counting Rules

Counting Rule 1

• counting the number of possible outcomes

• If there are k1 choices for the first event, k2 choices for the second event, k_n choices for the nth event, ►Then the total number of possible outcomes are (k1)(k2)(k3)…(kn)

Counting Rule 2 – Permutations

• n! = (n)(n – 1)…(1)

• The number of ways of arranging X objects selected from n objects in order is:

• 

Counting Rule 3 – Combinations

• The number of ways of selecting X objects from n objects, without regard to order, is