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### Intro to Probability

STA 220 – Lecture #5

Randomness and Probability

- We call a phenomenon if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions
- The of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions

Probability Models

- The description of a random phenomenon in the language of mathematics is called a
- A probability model consists of 2 parts:
- A list of
- A for each outcome

Probability Models

- Example: Toss a coin.
- We do not know
- But we do know:
- The outcome will be either heads or tails
- We believe that each of these outcomes has a probability of ½

Probability Models

- The of a random phenomenon is the set of all possible outcomes
- Example: Toss a coin
- S = {heads, tails} or S = {H, T}

- Example: Toss a coin 4 times. Count # of Heads
- S = {0,1,2,3,4}

- Example: Roll a die
- S =

Probability Models

- Example: Suppose that in conducting an opinion poll you select four people at random from a large population and ask each if he or she favors reducing federal spending on low-interest student loans. The answers are “Yes” or “No”. Interested in the number of “Yes” responses.
- S =

Intuitive Probability

- An is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.
- In a probability model, events have

Intuitive Probability

- Probability Rules
- The probability P(A) of any event A satisfies
- If S is the sample space in a probability model,
then P(S) =

3. Two events A and B are if they have no outcomes in common and so can never occur together. If A and B are disjoint,

P(A or B) =

4. The of any event A is the event that A does not occur, written as AC. The complement rule states that

P(AC) = 1 – P(A)

Intuitive Probability

- A picture that shows the sample space S as a rectangular area and events as areas within S is called a

S

A

B

Intuitive Probability

- Example
- Distance learning courses are rapidly gaining popularity among college students. Choose at random an undergraduate taking a distance learning course for credit, and record the student’s age. Here is the probability model:

Intuitive Probability

The probability that the student we draw is not in the traditional undergraduate age range of 18 and 23 years is, by the complement rule,

P(not 18 to 23 years) =

= 1 – 0.57

= 0.43

Intuitive Probability

The events “30 to 39 years” and “40 years or over” are disjoint because no student can be in both age groups. So the addition rule says:

P(not 30 years or over) =

= 0.14 + 0.12

= 0.26

Finite Sample Space

- Probabilities in a finite sample space
- Assign a probability to each individual outcome. These probabilities must be numbers between 0 and 1 and must have sum 1
- The probability of any event is the sum of

Finite Sample Space

- Example
- Faked numbers in tax returns, payment records, invoices, expense account claims, and many other settings often display patterns that aren’t present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as

Finite Sample Space

- Benford’s law

Benford’s law usually applies to the first digits of the sizes of similar quantities, such as invoices, expense account claims, and county populations. Investigators can detect fraud by comparing the first digits in records such as invoices paid by a business with these probabilities.

Finite Sample Space

- Consider the events
- A = (first digit is 1)
- B = (first digit is 6 or greater)

- From the table of probabilities,
- P(A) = P(1) =
- P(B) = P(6)+ P(7)+ P(8)+ P(9)
= = 0.222

Finite Sample Space

- The probability that a first digit is anything other than 1 is, by the complement rule,
P(Ac) = 1 – P(A)

= 1 – 0.301 =

- The events A and B are disjoint, so the probability that a first digit is either 1 or 6 or greater is, by the addition rule,
P(A or B) =

= 0.301 + 0.222 = 0.523

Finite Sample Space

- Be careful to apply the addition rule only to disjoint events. Check that the probability of the event C that a first digit is odd is
P(C) = P(1)+ P(3)+ P(5)+ P(7)+ P(9)=

- The probability
P(B or C) = P(1)+ P(3)+ P(5)+ P(6)+

P(7)+ P(8)+ P(9)=

is not the sum of the P(B) and P(C), because events B and C are not disjoint. Outcomes and are common to both events.

Equally likely outcomes

- In some circumstances, we are willing to assume that individual outcomes are equally likely because of some balance in the phenomenon
- Examples:
- Ordinary coins have a physical balance that should make heads and tails equally likely
- The table of random digits comes from a deliberate randomization

Equally likely outcomes

- Example
- You might think that first digits are distributed “at random” among the digits 1 to 9. The 9 possible outcomes would then be equally likely.
- The sample space for a single first digit is:
S =

- Because the total probability must be 1, the probability of each of the 9 outcomes must be

Equally likely outcomes

- The probability of the event B that a randomly chosen first digit is 6 or greater is
P(B) = P(6) + P(7) + P(8) + P(9)

=

= 4/9 = 0.444

Equally likely outcomes

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

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