probability part 1 – fundamental and factorial counting rules
TRANSCRIPT
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
Fundamental Counting Rule
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?
Fundamental Counting Rule
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.
Fundamental Counting Rule
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.
Fundamental Counting Rule
Variation for n events each occurring the same number of times (k) No. of ways = kn
Fundamental Counting Rule
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