PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH

Lesson 5PROBABILITY 

CORRECTIONS

1. Central tendency: Mean, Median & Mode

There are a few extreme scores in the distribution

Some scores have undetermined valuesThere are open ended distributionThe data measured on an ordinal scale

nominal

There are a few extreme scores in the distribution

Some scores have undetermined valuesThere are open ended distributionThe data measured on an ordinal scale

CORRECTIONS

MISCONCEPTION:

Many of you wrote: “for the sample A, the semi-interquartile range is more appropriate, because the semi-interquartile of sample A is smaller than that of sample B.”

Two samples are as follows: Sample A: 7, 9, 10, 8, 9, 12 Sample B: 13, 5, 9, 1, 17, 9

CORRECTIONS

2. Variability: Range, Semi-interquartile range, variance, standard deviation

1.Extreme scores.

2. Sample size.

3. Stability under sampling

4. Open-ended distributions

CORRECTIONS

Calculating sample standard deviation:

Population Sample

Mean µ X

variance σ2 = SS/N s2=SS/n-1

Standard deviation σ = √SS/N s = √SS/n-1

Interpretations of Probability

1. The frequency interpretation of probability

The probability that some specific outcome of a process will be obtained can be interpreted to mean the relative frequency with which that outcome would be obtained if the process were repeated a large number of times under similar conditions.

Interpretations of Probability

2. The classical interpretation of probability

It is based on the concept of equally likely outcomes.

Interpretations of Probability

3. The subjective interpretation of probabilityThe probability that a person assigns to a

possible outcome of some process represents her/his own judgment of the likelihood that the outcome will be obtained. This judgment will be based on each person’s beliefs or information about the process.

It is appropriate to speak of a certian person’s subjective probability , rather than to speak of the true probability of that outcome.

Experiments

An experiment is the process of making observation.

Ex:

a. A coin is tossed 10 times. The experimenter might want to determine the probability that at least four heads will be obtained.

b. In an experiment in which a sample of 1000 transistors is to be selected from a large shipment of similar items and each selected item is to be inspected, a person might want to determine the probability that not more than one of the selected transistors will be defective.

Sample space

A sample space is a set of points corresponding to all distinctly possible outcomes of an experiment.

Ex: For the die tossing experiment,

Sample point

A sample point is a point in a sample space.

Ex: For the die tossing experiment,

Descrete sample

A descrete sample space is one that contains a finite number or countable infinity of sample points.

Ex: A coin is tossed two times.

Event

For a descrete sample space, an event is any subset of it.

Ex: a. A coin is tossed two times.

b. For the die tossing experiment

Note: simple eventEx: observe a 6.

Summarizing example

Tossing a Coin: Suppose that a coin is tossed three times. Then

Experiment :

Sample space :

Sample point:

Events:

Simple event:

Definition of probability

Axiom 1. For every event A, Pr (A)≥0.

Axiom 2. Pr (S) = 1.

Axiom 3 For every infinite sequence of disjoint events

11

)Pr(Pri

ii

i AA

,....., 2AAi

11

)Pr(Pri

ii

i AA

,....., 2AAi

Theorem 1

Pr 0

Theorem 2

1 1

Pr Pr( )nn

i ii i

A A

,....., 2AAi

For every finite sequence of n disjoint events

Theorem 3

Pr( ) 1 Pr( )A A

For every event A

Theorem 4

Pr( ) Pr( )A B

If , thenA B

Theorem 5

0 Pr( ) 1A

For every event A,

Theorem 6

Pr( ) Pr( ) Pr( ) Pr( )A B A B A B

For every two events A and B,

Summarizing example

Diagnosing Diseases: A patient arrives at a doctor’s office with a sore throat and low grade fever. After an exam, the doctor decides that the patient has either a bacterial infection or a viral infection or both. The doctor decides that there is a probability of 0.7 that the patient has a bacterial infection and a probability of 0.4 that the person has a viral infection. What is the probability that the patient has both infection?

Summarizing example 2

Demands for Utilities: A contractor is building an office complex and needs to plan for water and electricity demands (sizes of pipes, conduit, and wires). After consulting with prospective tenants and examining historical data, the contractor decides that the demand for electricity will range between 1 million and 150 million kilowatt-hours per day and water demand will be between 4 and 200 (in thousand gallons per day). All combinations of ellectrical and water demand are considered possible.

Finite sample space

Experiments include a finite number of possible outcomes.

The number is the probability that the outcome of the experiment will be

1 2, ,......., nS s s s

ip, ( 1, 2,3,...., )is i n

1

0

1

i

n

ii

p

p

If the probability assigned to each of the outcomes is 1/n, then this sample space S is a simple sample space.

Summarizing example

Fiber breaks : consider an experiment in which five fibers having different lenghts are subjected to a testing process to learn which fiber will break first. Suppose that the lenghts of the five fibers are 1, 2, 3, 4, and 5 meters, respectively. Suppose also that probability that any given fiber will be the first to break is proportional to the lenght of that fiber. Determine the probability that the lenght of the fiber that breaks first is not more than 3 meters.

The probability of a union of events

If the events are disjoint,

Theorem: For every three events,

1 1

Pr Pr( )nn

i ii i

A A

1 2 3Pr( ) ...A A A

Summarizing example

Student Enrollment: Among a group of 200 students, 137 students are enrolled in a mathemtical class, 50 students are enrolled in a history class, and 124 students are enrolled in a music class. Furthermore, the number of students enrolled in both the mathematics and history classes is 33; the number enrolled in both the history and music class 29, and the number enrolled in both the methemtics and music class is 92. Finally, the number of students enrolled in all three classes is 18. Determine the probability that a student slected at random from the group of 200 stundents will be enrolled in at least one of the three classes.

Teaching probability

Constructing probability examples

Work with examples such as the probability of boy and girl births and use probability models of real outcomes.

These are more interesting and are known than card and crap games.

Teaching probability

Random numbers via dice or handouts

a. Rolling the dice ones gives a random digit.

b. If it is too inconvenient, you can prepare handouts of random numbers for your students.

c. You can use already existing material.

Ex: telephone book.

Teaching probability

Probability of compound events

Use “babies” or “real vs. fake coin flips”.

Babies: Students enjoy examples involving families and babies.

EX: We adapt a standard problem in probability by asking students which of the following sequences of boy and girl births is most likely, given that a family has four children: bbbb, bgbg, or gggg.

Teaching probability

Probability of compound eventsReal vs. fake coin flips: Students often have diffuculties with

probability of distributions.We pick two students to be “judges” and one to be the

“recorder” and divide the others in the class into two groups. One group is instructed to flip a coin 100 times, or flip 10 coins 10 times each, or follow some similarly defined protocol, and then to write the results, in order, on a sheet of paper, writing heads as “1” and tails as “0”.The second group is instructed to create a sequence of 100 “0”s ans “1”s that are intended to look like the result of coin flips- but they are to do this without flipping any coins or randomization device- and to write this sequence on a sheet of paper.

Teaching probability

Probability modelingWe can apply probabilty distributions to real

phenomena.Ex: Airplane failure (and other rare events)

Looking back historical data gave probability estimate of about 2%. Its deadly accident was calculated as 82%. what is the probabilty of that a person will be dead in an airplane accident due to airplane failure?