1

Topics that will be tested in

Quiz 2:

•Discrete Probability

•Binomial Probability

•Poisson Probability

•Normal Probability

Key Answers

Page 241

5.10) E(X)=$80,000

5.11a)

# years XP(X)

1$2.00 0.5

23.50 0.2

34.50 0.2

56.50 0.1

5.11b) $406.25

5.12a) Project 3

b) Project 3

5.13) Country C

Q5.14) Tire C

Q 5.15) Order 11 units

Binomial Probability Distribution

Which of the above probability

distributions are for a discrete

random variable? Continuous?

Discrete Continuous

Binomial Normal

Poisson Inverse

Normal

Central Limit

Theorem

Discrete

Random Variables

A discrete random variablediscrete random variable may assume either a finite number

of values or an infinite sequence of values.

FINITE

Number of values

INFINITE

Number of values

How to differentiate between

discrete and continuous random

variables?

DISCRETE

•The observation can be

counted

CONTINUOUS

•The observation can be

measured

ElTi(i

•Example: the number of

students who drive to

school

•What is the probability

that at most 110 students

will drive to school?

•

E

xamp

l

e:

Ti

me

(i

n

seconds) is measured and

not counted.

•What is the probability

that the download time for

a home page on a Web

browser is between 7 and

10 seconds?

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2

Discrete Random Variable

Examples: Finite/Infinite number

values

Experiment Random

Variable

Possible

Values

Make 100 sales calls

#Sales

0,1,2, ..., 100

Fi it

Make

100

sales

calls

#

Sales

0,

1,

2,

...,

100

Inspect 70 radios # Defective 0, 1, 2, ..., 70

Answer 33 questions # Correct0, 1, 2, ..., 33

Count cars at toll

between 11:00 & 1:00

# Cars

arriving

0, 1, 2, ..., f

Fi

n

it

e

Number

values

Infinite

Number

values

What is a Binomial Probability

Distribution?

Characteristics of the Binomial

Distribution

1. The experiment consists of a sequence of n

identical trials (repetitions).

2. Each trial has only one of the two possible mutually

exclusive outcomes, success or a failure. Two

outcomes, success and failure, are possible on each

trial.

3. The probability of each outcome does not change

from trial to trial.

4. The trials are independent, thus we must sample with

replacement.

The random variable that we are interested in is X,

Number of ‘successes’ in a sample

of nobservations (trials)

X= # of successes

Summary: Three pieces of info

1. Sequence of nidentical trials / A sample of ‘n’

items are selected from a large population

2. Constant Probability (3) for each Trial

–

e.g. Probability of getting atail is the sameeach

e.g.

Probability

of

getting

a

tail

is

the

same

each

time we toss the coin and each light bulb has the

same probability of being defective

3. Each trial has 2 outcomes

–‘Success’ (desired outcome) or ‘failure’

–X= # of successes

To calculate the probability that X takes on a

specific value we use the Binomial Probability

Distribution Function

What is the Binomial Probability

Distribution Function?

xnx

xnx

n

xXP

)1(

)!(!

!

)(

SS

P(X=x) = probability that x successes given a knowledge of

n and 3

X= number of ‘successes’ in

sample, (X= 0, 1, 2, ..., n)

S= probability of each ‘success’

n= sample size

Use the CASIO calculator to obtain the

Binomial Probability

See Handout on “CASIO Calculator –Lesson 3 :

Binomial and Poisson – page 250

STAT F5(DIST) F5(BINM) F1(Bpd)

Using the CASIO calculator

Then select the following options

You will see: Binomial P.D

Data : Variable

X :

Numtrial :

P :

Save Res: None

Execute

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3

Bpd and Bcd functions

The Casio calculator has 2 Binomial Probability

functions programmed into its memory.

These are:

1) Bpd stands for Binomial Probability Distribution.

This function calculators a Binomial Probability of the form:

P( X#)

P(

X

=

#)

2) Bcd stands for Binomial Cumulative Distribution.

This function calculators a Binomial Probability of the form:

P( X #)

Use the CASIO calculator to obtain the

Binomial Probability

•See Handout on “CASIO Calculator –

Lesson 3 : Binomial and Poisson – page 250

X

=

# oftails

STATF5(DIST) F5(BINM) F1(Bpd)

Experiment: Toss 1 coin 6 times in a row..

What’s the probability of getting 2 tails?

X

#

of

tails

n = 6

3= 0.5

STAT

F5(DIST)

F5(BINM)

F1(Bpd)

Then select the following options

You will see:

Binomial P.D

Data : Variable

x : 0

Numtrial : 0

P : 0

Save Res: None

Execute

2 EXE

6 EXE

0.5 EXE

F2 (CALC)

Presentation of your Solution

(for Binomial Probability)

BINOMIAL TEMPLATE

Your answers should look like the following template:

X = # of _____________________

(successes)

n = __________________

= _________________

(decimals)

P ( ) = P ( X symbol # ) = Bpd ( ) = 0.xxxx

(words) ( <, >, , , =) or Bcd ( ) 4 decimals

If the question does not directly say what the number is.

Example 5.11

A survey of people in the 30 and under 40 age bracket shows

that 43% of them have investments in mutual funds. In a

particular condominium there are 18 adults in this age

bracket. What is the probability that:

1. 5 of them will have investments in

mutual funds?

mutual

funds?

2.At most 2 of these people will have investments in mutual

funds?

3.More than 10 of these people will have investments in

mutual funds?

4.From 3 to 9 of these people will have investments in mutual

funds?

1) What is the probability that 5 of them will

have investments in mutual funds?

How do you write the probability?

P(X=5)

Define X:

What is X? What are you observing or counting???

X= # of people in their 30’s with investments in mutual

funds.

Next, state n and

Where do you obtain the values of n and ?

A survey of people in the 30 and under 40 age bracket shows

that 43% of them have investments in mutual funds.

In a particular condominium there are 18 adults in this age bracket.

1) What is the probability that 5 of them will

have investments in mutual funds?

X= # of people in their 30’s with investments in mutual

funds.

n

=

18

n

18

= 0.43 = 43%WRONG!

P(X=5) = Bpd (5, 18, 0.43) = 0.0844

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