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.

abcdeabcdeabcdeabcde TFTF Tree Diagram of the events T & a T & b T & c T & d T & e F & a F & b F & c F & d F & 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)

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

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: In the horse race example above we would write : (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) n 1 !. n 2 ! n k ! n!n! If the preceding requirements are satisfied, and if there are n 1 alike, n 2 alike,... n k 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! n C r = 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

permutation When different orderings of the same items are to be counted separately, we have a permutation problem, combination but when different orderings are not to be counted separately, we have a combination 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 7 Elected 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

{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