Chapter 9

• Introducing Probability

- A bridge from Descriptive Statistics to Inferential Statistics

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

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 don’t know for sure what will happen. For example, when we toss a coin once, we don’t know exactly what we will get (Head or Tail).

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.

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.

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.

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”?

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)}

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”?

Probability rules• Rule 1: For any event E, 0<=P(E)<=1.• 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. Let’s say the event E an even number is thrown and F an odd number is thrown.

• Question: Are E and F disjoint?

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.

Example 9.7 (P.233)

• S={1,2,3,4,5,6,7,8,9}• Let X=first 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(X>=6)=?• P(X>6)=?• P(5<X<9)=?

Assigning probabilities:

• Case II: intervals of outcomes

• Example: P(0.3<=Y<=0.7) =?– Y = a random number between 0 and 1– S={all numbers between 0 and 1} = [0,1]

• Idea: area under a density curve.

• Example 9.8 (page 235)

• Exercise 9.9 (page 237)

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

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.

Random variables

• Example: tossing a coin 4 times– S={HHHH, HHHT,HHTH,…,TTTT}, It has

16 possible outcomes. – Suppose that we are interested in number of

heads, then S={0,1,2,3,4}– We can assign probabilities to each outcome.

• Example: Uniform distribution over [0,1]– S=(0,1)– We can assign probabilities over interval