Introducing Probability

1
Chapter 9

• Introducing Probability
• - A bridge from Descriptive Statistics to
  Inferential Statistics

2
Chapter outline

• The idea of probability
• Thinking about the randomness
• Probability models
• Assigning probabilities finite number of
  outcomes
• Assigning probabilities intervals of outcomes
• Normal probability models
• Random variables

3
The idea of probability

• Some event where the outcomes is uncertain.
  Examples of such outcomes would be the roll of a
  die, the amount of rain that we get tomorrow, or
  who will be the president of the United Sates in
  the year 2004.
• In each case, we dont know for sure what will
  happen. For example, when we toss a coin once, we
  dont know exactly what we will get (Head or
  Tail).

4
The idea of probability

• Probability theory allows us to make some sense
  out of happening due to chance.
• Example If you flip a coin many times, about
  half the time you get heads and the other half
  you get tails. In general, the more times you
  flip the coin, the closer the ratio of heads to
  tails comes to one.
• Question Why should this always be so?
• Answer There is a mathematical rule governing
  coin flipping it says that when you flip a
  coin, the outcomes are about even between heads
  and tails.

5
Thinking about randomness

• A phenomenon is random if each outcome is
  uncertain but there is nonetheless a regular
  distribution of outcomes in a large number of
  repetitions.
• Examples of random phenomena
• The probability of any outcomes of a random
  phenomenon is the proportion of times the outcome
  would occur in a very long series of repetitions.

6
Definitions

• Sample space the set of all possible outcomes.
  We denote S
• Event an outcome or a set of outcomes of a
  random phenomenon. An event is a subset of the
  sample space.
• Probability is the proportion of success of an
  event.
• Probability model a mathematical description of
  a random phenomenon consisting of two parts S
  and a way of assigning probabilities to events.

7
Example 9.6 (P.232)

• We roll two dice and record the up-faces in order
  (first die, second die)
• What is the sample space S?
• What is the event A roll a 5?

8
Probability models

• Example 9.6 (p.232) Rolling two dice
• We roll two dice and record the up-faces in order
  (first die, second die)
• All possible outcomes
• (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
• (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
• (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
• (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
• (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
• (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
• Roll a 5 (1,4) (2,3) (3,2) (4,1)

9
Example 9.4 (P.229)

• We roll two dice and count the spots on the
  up-faces.
• What is the sample space S?
• What is the event B I get an even number.?
• What is the event C I get an odd number. ?
• What is the event D I get a count less than
  4?

10
Probability rules

• Rule 1 For any event E, 0ltP(E)lt1.
• Rule 2 If S is the sample space in a probability
  model, then P(S)1.
• Rule 3 For any event E, P(E does not occur)
• 1-P(E occurs)
• Rule 4 For two disjoint (mutually exclusive)
  events E and F, P(E or F) P(E) P(F)
• In a probability experiment, two events E and F
  are said to be disjoint if they cannot both occur
  simultaneously.
• For example we throw a die once. Lets say
  the event E an even number is thrown and F an odd
  number is thrown.
• Question Are E and F disjoint?

11
Assigning probabilities

• Case I finite number of outcomes
• Assign a probability to each individual outcome.
• These probabilities must be numbers between 0 and
  1 and must have sum 1.
• Probability histogram is useful.

12
Example 9.7 (P.233)

• S1,2,3,4,5,6,7,8,9
• Let Xfirst digit.
• Probability model
• X 1 2 3 4 5 6 7
  8 9
• P 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9
  1/9
• P(Xgt6)?
• P(Xgt6)?
• P(5ltXlt9)?

13
Assigning probabilities

• Case II intervals of outcomes
• Example P(0.3ltYlt0.7) ?
• Y a random number between 0 and 1
• Sall numbers between 0 and 1 0,1
• Idea area under a density curve.

14

• Example 9.8 (page 235)
• Exercise 9.9 (page 237)

15
Random variables

• Random variable a variable whose value is a
  numerical outcome of a random phenomenon. There
  are two kinds of random variables corresponding
  to the ways of assigning probabilities.
• Discrete random variable spread on the number
  line discretely.
• Continuous random variable interval

16
Probability distributions

• Probability distribution of a random variable X
  it tells what values X can take and how to assign
  probabilities to those values.
• Probability of discrete random variable list of
  the possible value of X and their probabilities
• Probability of continuous random variable
  density curve.

17
Random variables

• Example tossing a coin 4 times
• SHHHH, HHHT,HHTH,,TTTT, It has 16 possible
  outcomes.
• Suppose that we are interested in number of
  heads, then S0,1,2,3,4
• We can assign probabilities to each outcome.
• Example Uniform distribution over 0,1
• S(0,1)
• We can assign probabilities over interval