Probability Part 1 – Fundamental and Factorial Counting Rules.

Probability

Part 1 – Fundamental and Factorial Counting Rules

Probability

Warm-up – 1) 2/3 x 6/7 = 2) 54/64 = 3) 4/13 + ¼ + ½ - 7/12 =

Probability

Agenda Warm-up Objective – To introduce the concept of

probability and the counting rules Summary Homework

Idea of Probability

Probability is the science of chance behavior

Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run this is why we can use probability to gain

useful results from random samples and randomized comparative experiments

Probability

There are two ways of looking at probability: Random Chance Probability- Events seem to

happen in a random way Subjective Probability – Skill of the “player”

effects the outcome

Probability History

During the mid-1600s,a professional gambler named Chevalier de Méré made a considerable amount of money on a gambling game. He would bet unsuspecting patrons that in four rolls of a die, he could get at least one 6. He was so successful at the game that some people refused to play. He decided that a new game was necessary to continue his winnings. By reasoning, he figured he could roll at least one double 6 in 24 rolls of two dice, but his reasoning was incorrect and he lost systematically.

Unable to figure out why, he contacted a mathematician named Blaise Pascal (1623–1662) to find out why. Pascal became interested and began studying probability theory.

He corresponded with a French government official, Pierre de Fermat (1601–1665), whose hobby was mathematics. Together the two formulated the beginnings of probability theory.

Outcomes

Before calculating probability- Need to understand the number of ways an

event can occur This involves counting rules.

Probability

A probability experimentprobability experiment is a process that leads to well-defined results called outcomes.

An outcomeoutcome is the result of a single trial of a probability experiment.

A sample space is the set of all possible outcomes of a probability experiment.

Tree Diagrams

A tree diagramtree diagram is a device used to list all possibilities of a sequence of events in a systematic way.

Tree Diagrams - - Example

Suppose a sales person can travel from Boston to New York and from New York to Philadelphia by plane, train, or automobile. Display the information using a tree diagram.

Tree Diagrams - - Example

Philadelphia

Boston

New York

Plane

Train

Auto

Auto

Train

Plane

Train

Auto

Plane

Auto

Train

Plane

Plane, Auto

Plane, Train

Plane, PlaneTrain, Auto

Train, Train

Train, PlaneAuto,AutoAuto, Train

Auto, Plane

Tree Diagram

Thus – by counting the number of stems at the end of the tree – you determine the number of different ways the sequence of events can occur.

Fundamental Counting Rule

We can also calculate the number of ways by using:

Fundamental Counting Rule :Fundamental Counting Rule : In a sequence of nn events in which the first one has kk11 possibilities and the second event has kk22 and the third has kk33, and so forth, the total possibilities of the sequence will be kk11kk22kk33kknn or kii= 1

n

Tree Diagrams

Example – You have 3 pants, 4 t-shirts, 3 hats, and 2

sneakers. Using a tree diagram, how many different outfits can be created?

Example You have 3 pants, 4 t-shirts, 3 hats, and 2

sneakers. Using the fundamental counting rule, how many different outfits can be created?

3 x 4 x 3 x 2 = 72


Employees of SHS are to be issued special coded identification cards. The card consists of 4 letters of the alphabet. Each letter can be used up to 4 times in the code. How many different ID cards can be issued?


Since 4 letters are to be used, there are 4 spaces to fill ( _ _ _ _ ). Since there are 26 different letters to select from and each letter can be used up to 4 times, then the total number of identification cards that can be made is 26 2626 26= 456,976.


The digits 0, 1, 2, 3, and 4 are to be used in a 4-digit ID card. How many different cards are possible if repetitions are permitted?

Solution:Solution: Since there are four spaces to fill and five choices for each space, the solution is 5 5 5 5 = 54 = 625.


Variation for n events each occurring the same number of times (k) No. of ways = kn


1) Create a tree diagram to illustrate going from home to school to work and back home by either walking, by bike, or by car. How many different ways can you make the trip?

2) If you had 3 math books, 4 English books, 2 social studies, and 3 science books. How many different ways could they be arranged on your shelf.

3) A license plate in MA contains 6 characters – 4 numbers followed by 2 letters. Assuming repeats are possible, how many different license plates are possible?

Answers

Problem 1

Home

Car

Bike

Walk

School

Car

Bike

Walk

Car

Bike

Walk

Car

Bike

Walk

Work Home

Car

Bike

WalkCar

Bike

WalkCar

Bike

Walk

Car

Bike

WalkCar

Bike

WalkCar

Bike

Walk

Car

Bike

WalkCar

Bike

WalkCar

Bike

Walk

Answers

Problem 2 3 x 4 x 2 x 6 = 144 possible arrangements

Problem 3 10 x10 x 10 x 10 x 26 x 26= 6,760,000 license

plates

Counting Rules

Summary Tree Diagrams Fundamental Counting Rule Factorial Counting Rule

Counting Rules

Homework Handout #1

Probability Part 1 – Fundamental and Factorial Counting Rules.

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