Introduction to Probability

1
Introduction to Probability

• The classical definition of probability is
  identified with the works of Pierre Simon
  Laplace. As stated in his Théorie analytique des
  probabilités.
• likelihood of a given event's occurrence, which
  is expressed as a number between 1 and 0.
• In general, the probability of an event is the
  number of favorable outcomes divided by the total
  number of possible outcomes.

2
Components of Probability

• A random experiment
• Possible outcomes of that experiment
• An event of interest

3
Random experiment

• An experiment that has a number of possible
  outcomes
• Time sequence
• Rolling dice/ sex of infants/ coin tossing

4
Two remarks

• Exclusive one outcome preclude another
• Equally likely outcomes equally probable
  outcomes

5
Four types of events

• Sure event
• Impossible event
• negation event
• Composite event

6
Conditional probability

• what is the probability that the total of two
  dice will be greater than 8 given that the first
  die is a 6?
• p(AB) is the probability of event A given that
  event B occurred the expression to the left of
  the vertical bar represents the event and the
  expression to the right of the vertical bar
  represents the condition.

7
Probability of A and B

• If A and B are Independent
• If A and B are Not Independent

8
Binomial distribution

• An event with two mutually exclusive possible
  outcomes
• Success Failure
• where P(r) is the probability of exactly r
  successes, N is the number of events, and p is
  the probability of success on any one trial.

9
Event Characteristics

• (a) are dichotomous (fall into only two
  categories)
• (b) are mutually exclusive
• (c) are independent and
• (d) are randomly selected

10
What is a normal distribution?
11
Standard normal distribution

• Normal distribution with a mean of 0 and a
  standard deviation of 1
• Z distribution

12
Areas under portions of the standard normal
distribution

• About .68 (.34 .34) of the distribution is
  between -1 and 1 while about .96 of the
  distribution is between -2 and 2.

13
What's so important about the normal
distribution?

• Many psychological and educational variables are
  distributed approximately normally
• Many kinds of statistical tests can be derived
  for normal distributions
• if the mean and standard deviation of a normal
  distribution are known, it is easy to convert
  back and forth from raw scores to percentiles.

14
Converting to percentiles and back
15
Area under a portion of the normal curve
16
Overview of point estimation

• A single number or a range of scores
• Point estimate or Interval estimate
• a point estimate of the standard deviation is
  used in the calculation of a confidence interval
  for µ. Point estimates of parameters are often
  used in the formulas for significance testing.

17
Characteristics of Estimators

• Unbiased ness
• Consistency
• Relative efficiency