IT College

Introduction to Computer Statistical Packages

Lecture 9 Eng. Heba Hamad

2010

Chapter 4 (part 3) Probability

Counting

Key Concept

In many probability problems, the big obstacle is finding the total number of outcomes, and this section presents several methods for finding such numbers without directly listing and counting the possibilities.

Content

In this section we will discuss the counting rule: The fundamental counting rule.

The permutations rule (when items are all different). The permutations rule (when some items are identical to others). The combinations rule.

The factorial rule.

Fundamental Counting Rule

For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m n ways.

a

b

c

d

e

a

b

c

d

e

T

F

Tree Diagram of the eventsT & aT & bT & cT & dT & eF & aF & bF & cF & dF & e

m = 2 n = 5 m*n = 10

Examples

Breaking Code …( two-character code consisting of a letter followed by a digit)

Computer Design…(byte is defined to be 8 bits)

Safety in Numbers…(Electronic Key) Chances and Skills …(someone claims

that he has ability to roll a die in such a way that 6 will almost always occur, how many outcomes are possible with five rolls of a die)

Factorial Rule

Examples

Survey Questions…(5 questions) Air Routes…(How many routes are possible?)

Routes to All 50 Capitals …(How many different routes are possible?...if you want to calculate the shortest possible route)

6123!3

A collection of n different items can be arranged in order n! different ways.

(This factorial rule reflects the fact that the first item may be selected in n different ways, the second item may be selected in

n – 1 ways, and so on.)

Factorial Rule

Notation

The factorial symbol ! denotes the product of decreasing positive whole numbers.

For example,

By special definition, 0! = 1

241234!4

10! 10 9 8 7 6 5 4 3 2 1 3,628,800

Permutations Rule(when items are all different)

Examples

Routes to 4 Capitals of the 50…(you do not have time to travel to 50 capitals, How many different routes are possible)

Novels of a n author…(you have to chose the best three novels of ten novels for prizes , How many different outcomes are possible)

Permutations Rule(when items are all different)

)n - r!(n r

P = n!

Then number of permutations (or sequences) of r items selected from n available items (without replacement) is

Requirements:

1. There are n different items available. (This rule does not apply if some of the items are identical to others.)

2. We select r of the n items (without replacement).

3. We consider rearrangements of the same items to be different sequences. (The permutation of ABC is different from CBA and is counted separately.)

Example A horse race has 14 horses, how many different possible ways can the top 3 horses finish? There are 14 possibilities for which horse finishes first, 13 for second, and 12 for third, so by the Fundamental Counting Principle, there are:

14 13 12 2184

In the horse race example above we would write :

14 3 2184P

)14 - 3!(

14 3

P =14!

Permutations Rule)when some items are identical to

others(

Examples

Letters Arrangements…(How many different ways, you can arrange the letters of the word Statistics)

Diet and Regular Colas…(you have 4 diet and 5 regular colas , How many ways can we arrange the letters DDDDRRRRR?)

Permutations Rule)when some items are identical to

others(

n1! . n2! .. . . . . . . nk!

n!

If the preceding requirements are satisfied, and if there are n1 alike, n2 alike, . . . nk alike, the number of permutations (or sequences) of all items selected without replacement is

Requirements:

1. There are n items available, and some items are identical to others.

2. We select all of the n items (without replacement).

3. We consider rearrangements of distinct items to be different sequences.

Combinations Rule

Examples

Elected Offices…(From 9 members in the college, elect 3 persons committee to oversee buildings and grounds How many different 3 persons committee are possible)

(n - r ) !r!

n!nCr =

Combinations Rule

If the preceding requirements are satisfied, the number of combinations of r items selected from n different items is

Requirements:

1. There are n different items available.

2. We select r of the n items (without replacement).

3. We consider rearrangements of the same items to be the same. (The combination of ABC is the same as CBA.)

Comparison between Combinations Rule and

Permutations Rule

When different orderings of the same items are to be counted separately, we have a permutationpermutation

problem ,

but when different orderings are not to be counted separately, we have a combinationcombination problem.

Permutations versus Combinations

Examples

Elected Offices…(From 9 members in the college, elect 3 persons committee to oversee buildings and grounds How many different 3 persons committee are possible)

Elected Offices…(From 9 members in the college, elect 3 persons, one is the chairperson, vice chairperson and secretary)

Example 7Elected Offices:From 9 member, each year, they elect 3-person committee to oversee( Chairperson, vice chairperson , secretary)

a)how many different 3-person committees are possible?

b) how many different slates of candidates are possible?

Possible permutations of three letters from the collection of five letters

abc abd abe acd ace ade bcd bce bde cde

acb adb aeb adc aec aed bdc bec bed ced

bac bad bae cad cae dae cbd cbe dbe dce

bca bda bea cda cea dea cdb ceb deb dec

cab dab eab dac eac ead dbc ebc ebd ecd

cba dba eba dca eca eda dcb ecb edb edc

{a,b,c} {a,b,d} {a,b,e} {a,c,d} {a,c,e}

{a,d,e} {b,c,d} {b,c,e} {b,d,e} {c,d,e}

Possible Combinations of three letters from the collection of

five letters