Section 5.1

What is Probability?

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Probability

• Probability is a numerical measurement of likelihood of an event.

• The probability of any event is a number between zero and one.

• Events with probability close to one are more likely to occur.

• Events with probability close to zero are less likely to occur.

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3

Probability• Probabilities are assigned

values from 0 to 1.

• The closer the probability of a given event is to 1, the more likely it is that the event will occur.

• The closer the probability of a given event is to 0, the less likely that the event will occur.

Probability Notation

If A represents an event,P(A)

represents the probability of A.

If P(A) = 1 Event A is certain to occur

If P(A) = 0 Event A is certain not to occur

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Three methods to find probabilities:

• Intuition

• Relative frequency

• Equally likely outcomes

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Intuition Method of Determining Probability

• Incorporates past experience, judgment, or opinion.

• Is based upon level of confidence in the result• Example: “I am 95% sure that I will attend the

party.”

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Probability of an event = the fraction of the time that the event occurred

in the past = f n

where f = frequency of an eventn = sample size of n observations

Probability as Relative Frequency

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Example of Probability as Relative Frequency

If you note that 57 of the last 100 applicants for a job have been female, the probability that the next applicant is

female would be:

100

57

Equally Likely Outcomes

• No one result is expected to occur more frequently than any other.

When outcomes are equally like

Probability of an eventNumber of outcomes favorable to event

Total number of outcomes

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Example of Equally Likely Outcome Method

When rolling a die, the probability of getting a number less than three =

3

1

6

2

Example

A die is rolled once. The sample space is S = {1,2,3,4,5,6} Find the probability of rolling

a. a 3 b. an even numberSolution to a: P(3) = number of outcomes favor to 3 = 1/6 total number of outcomes

Solution to b:Rolling an even number describes the event {2,4,6}.This event can occur in 3 ways:P(even number) = number of outcomes favor to even number = total number of outcomes

3/6 = 1/2 5.1 / 11

Example

• You are dealt one card from a standard 52-card deck. Find the probability of being dealt a

a. KingSolution: P(King) = number of outcomes favor a king = total number of outcomes b. HeartSolution:P(heart) = number of outcomes favor to a heart = total number of outcomes

4 1

52 13

13 1

52 4

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Law of Large Numbers

In the long run, as the sample size increases and increases, the relative frequencies of outcomes get closer and closer to the

theoretical (or actual) probability value.

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Statistical Experiment or Statistical ObservationIs any random activity that results in a definite

outcomeEvent Is a collection of one or more outcomes of a

statistical experiment or observationSimple EventIs an outcome of a statistical experiment that

consists of one and only one of the outcomes of the experiment

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Sample Space• Is the set of all possible distinct outcomes of

an experiment• The sum of all probabilities of all simple

events in a sample space must equal one.

• Example: Sample Space for the rolling of an ordinary die: 1, 2, 3, 4, 5, 6

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For the experiment of rolling an ordinary die:

• P(even number) =

• P(result less than five) =

• P(not getting a two) =

2

1

6

3

3

2

6

4

6

5

Complement of Event A

• the event that A does not occur

• Notation for the complement of event A:

cA

Event A and its complement Ac

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Probability of a ComplementP(event A does not occur) = P(Ac) = 1 – P(A)So, P(A) + P(Ac) = 1

If the probability that it will snow today is 30%,

P(It will not snow) = 1 – P(snow) =

1 – 0.30 = 0.70

Probability of a Complement

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Probability Related to Statistics

• Probability makes statements about what will occur when samples are drawn from a known population.

• Statistics describes how samples are to be obtained and how inferences are to be made about unknown populations.

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Generating Random Numbers (calc)Random numbers can be quickly and easily generated by using the

graphing calculator.Let's start our investigation by looking at generating random integers.(The TI-84+ is being used on this page.)

Generating Random Integers on the Home Screen: (good for games) Go to MATH → PRB Choose #5 randInt(

From the home screen, enter the smallest value needed, followed by the largest value. Hitting ENTER will generate the random integers.(Random values may repeat.)This example generates random numbers from 1 to 25 (good for Bingo).

Adding a third parameter indicates the number of random integers that will appear on the screen at one time. 5.1 / 21

Generating Random Numbers (calc)Generating Random Integers in Lists: (good for statistical studies)

Go to MATH → PRB Choose #5 randInt(From the home screen, enter the randInt followed by the smallest value in

the desired range, the largest value, and the number of terms needed. The results are stored (STO) into List 1.This example stores 100 random integers from 0 to 1 in L1 tosimulate the toss of a coin.

OR, from the list screen, arrow up onto L2, and type randInt(0,1,100). Hit ENTER.Be sure to enter the third parameter so the calculator will know "how many" numbers to place in the list.

Such lists can be used to simulate the toss of one (or more) fair coin(s). The number of entries represents the number of tosses. An even random number represents heads, while an odd number represents tails.

If tossing one coin, use sum command to count the number of heads,where heads are 1, and tails are 0. 2nd STAT - MATH - #5 sum 5.1 / 22

Generating Random Numbers (calc)Using the rand command: (not integers)

The rand (MATH → PRB #1 rand) command will generate random values, not integers.

MATH → PRB #1 rand

The rand command will create a random number between 0 and 1. To generate a random number between 0 and 15, enterrand*15.

This last entry shows how to generate a list of 10 random numbers between 0 and 15 and store them in List 1.

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Generating Random Numbers (calc)Re-Seeding the Random Number Generator:

Calculators (and computers) are not capable of creating "truly random" numbers. They create what are called "pseudo-random" numbers, meaning they use a formula to create the values. To engage this formula, the calculator uses a starting value, called a "seed", and then creates the random numbers based upon this seed. If two calculators start with the same seed value, they will generate the same sequence of random values.

If you wish, you can control the starting "seed" value.To seed the random generator,choose a seed value and storeit into the rand command.

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Generating Random Numbers (calc)

Now, start generating your random values.If you feed two calculators the same seed value, they will each produce the same result when rand is entered. After running a RESET (DEFAULTS) the calculator will return to using its default seed value. Engaging rand will always produce the value seen above. This same value will appear on all TI-84+ calculators after a reset. If you wish to ensure that each student in the class has a different set of random numbers, assign a different number to each student as their seed value. You could also have the students enter their birth date as the random seed (04021990), assuming no two students have the same birth date.

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