Introduction to Research Design and Statistics

Probability


Random Phenomenon / Random Process

When individual outcomes are uncertain but there are nonetheless regular distributions of relative frequencies over a large number of trials.

Probability

In intuitive terms, long-term relative frequency

Probability Distribution

A set of probabilities associated with all possible outcomes of a variable. Probability distributions are abstract mathematical models which are useful to the extent to which they reflect distributions of variables in the real world.

Probability Models

Sample Space

Random Variable

A random variable is a numerial outcome of a random process whose exact value cannot be predicted exactly beforehand.

Event

Example:

Tossing three coins.

[{TTT} {TTH} {THT} {HTT} {THH} {HTH} {HHT} {HHH}]

Each of the above is an event.
All of the above, taken together, constitute the sample space.

Probability Rule #1

0<=P(A)<=1

Probability Rule #2

P(S) = 1

Equally Likely Outcomes

If a random phenomenon has k possible outcomes, all equally likely, than each individual outcome has the probability of 1/k. The probability of any event A is

Disjoint Events
(Mutually Exclusive Events)

Events that have no outcomes in common.

Venn Diagram

A picture used to make probability incomprehensible.

A and B are disjoint

Probability Rule #3
Addition Rule for Disjoint Events

If event A and B are disjoint,

Example:

In tossing 3 coins what is the probability that there will be all heads or no heads?

[{TTT} {TTH} {THT} {HTT} {THH} {HTH} {HHT} {HHH}]

P(A)=1/8 P(B)=1/8

P(A) + P(B) = 2/8 = 1/4

Complement of an Event

Another Venn Diagram

Probability Rule #4
The Complement Rule

For any event A, the probability of ~A is

Example:

In tossing 3 coins, what is the probability that there will not be exactly one head.

Event A is exactly one head.

[{TTT} {TTH} {THT} {HTT} {THH} {HTH} {HHT} {HHH}]

P(A) = 3/8
P(~A) = 1 - 3/8 = 5/8

Independent Events

Events A and B are independent if knowledge about the outcome of A does not change the probability of B.

Example:

Probability Rule #5
Multiplication Rule for Independent Events

If A and B are independent,

Example:

Pick a card at random from a deck of cards.

What is the probability that the card will be black and an ace.

Event A is card is black; Event B is card is an Ace
P(A) = 26/52 P(B) = 4/52
P(A and B) = 104/2704 = 2/52

How Las Vegas Makes Money

Random Variables

Random Variable

A variable whose value is a numerical outcome of a random phenomenon.

Discrete Random Variables

Probability Distribution

The set of probabilities assigned to a random variable X.

Example:

Number of Heads in Tossing 3 Coins
#Heads          Prob
  xi             pi
  0             1/8     
  1             3/8     
  2             3/8     
  3             1/8

Probability Histogram

A graphical representation of the probability distribution of a discrete random variable.

Example:

Number of Heads in Tossing 3 Coins

Continuous Random Variable

Example:

Normal Distributions

Means & Variances of Random Variables

Expected Value

Mean

Discrete Random Variable

mx = E(X) = Σxipi

Example:

Number of heads in tossing three coins.
#Heads     Prob     Product
    xi      pi
    0      1/8         0
    1      3/8        3/8
    2      3/8        6/8
    3      1/8        3/8
               Sum = 12/8 = 1.5 = E(X) = μx

Law of Large Numbers

The observed mean outcome in many independent trials must approach the mean of the probability distribution.

Rule #1 for Means

When Y = a + bX

Rule #2 for Means

When Z = X + Y

Variance

Discrete Random Variable

Example:

Number of Heads in Tossing 3 Coins

#Heads     (xi-μ)2      Prob     Product
   0         2.25       1/8        0.28
   1         0.25       3/8        0.09
   2         0.25       3/8        0.09
   3         2.25       1/8        0.28
                  Sum = Variance = 0.74
              Standard Deviation = 0.87

Rule #1 for Variances

When Y = a + bX

Rule #2 for Variances

When Z = X + Y and X & Y are independent

Rule #2a for Variances

When Z = X - Y and X & Y are independent


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Phil Ender, 18Nov99