Let’s work on some Let’s work on some definitions definitions Experiment- Experiment- is a situation is a situation involving chance that leads involving chance that leads to results called outcomes. to results called outcomes. An An outcome outcome is the result of a is the result of a single trial of an experiment single trial of an experiment An An event event is one or more is one or more outcomes of an experiment. outcomes of an experiment. Probability Probability is the measure of is the measure of how likely an event is. how likely an event is.
Transcript
Let’s work on some definitionsLet’s work on some definitions
Experiment-Experiment- is a situation involving is a situation involving chance that leads to results called chance that leads to results called outcomes.outcomes.
AnAn outcomeoutcome is the result of a single is the result of a single trial of an experimenttrial of an experiment
AnAn eventevent is one or more outcomes of is one or more outcomes of an experiment.an experiment.
ProbabilityProbability is the measure of how is the measure of how likely an event is.likely an event is.
Probability of an eventProbability of an event
The probability of event A is the The probability of event A is the number of ways event A can occur number of ways event A can occur divided by the total number of possible divided by the total number of possible outcomes.outcomes.
P(A)= P(A)= The # of ways an event can occurThe # of ways an event can occur
Total number of possible outcomesTotal number of possible outcomes
If P = 0, then the event _______ occur.
ProbabilityProbability
If P = 1, then the event _____ occur.
It is ________
It is ______
So probability is always a number between ____ and ____.
impossible
cannot
certain
must
10
All of the probabilities must add up to 100% or 1.0 in decimal form.
ComplementsComplements
Example: Classroom
P (picking a boy) = 0.60
P (picking a girl) = ____0.40
A glass jar containsA glass jar contains 6 red6 red, , 5 green5 green, , 8 blue8 blue andand 3 yellow3 yellow marbles.marbles.
Experiment: A marble chosen at random.Experiment: A marble chosen at random.
Possible outcomes: choosing aPossible outcomes: choosing a redred,, blueblue, , greengreen oror yellowyellow marble.marble.
Probabilities:Probabilities:
P(P(redred)) = = # of ways to choose red # of ways to choose red = = 6 6 = = 3 3
total number of marbles 22 11total number of marbles 22 11
You roll a six-sided die whose sides are numbered from 1 through 6. What is the probability of rolling an ODD number?
Ex. Ex.
P 12
=36
=
Tree Diagrams• Tree diagrams allow
us to see all possible outcomes of an event and calculate their probabilities.
• This tree diagram shows the probabilities of results of flipping three coins.
Use an appropriate method to find the number of outcomes in each of the following situations:
1. Your school cafeteria offers chicken or tuna sandwiches; chips or fruit; and milk, apple juice, or orange juice. If you purchase one sandwich, one side item and one drink, how many different lunches can you choose?
Sandwich(2) Side Item(2) Drink(3) Outcomes
chicken
tuna
There are 12 possible lunches.
chips
fruit
chips
fruit
apple juice orange juice milkapple juice orange juice milk
apple juice orange juice milkapple juice orange juice milk
chicken, chips, apple chicken, chips, orange chicken, chips, milkchicken, fruit, apple chicken, fruit, orange chicken, fruit, milk
tuna, chips, apple tuna, chips, orange tuna, chips, milktuna, fruit, apple tuna, fruit, orange tuna, fruit, milk
Multiplication Counting Principle(aka Fundamental Counting Principle)
• At a sporting goods store, skateboards are available in 8 different deck designs. Each deck design is available with 4 different wheel assemblies. How many skateboard choices does the store offer?
32
Multiplication Counting Principle• A father takes his son Tanner to Wendy’s
for lunch. He tells Tanner he can get the 5 piece nuggets, a spicy chicken sandwich, or a single for the main entrée. For sides: he can get fries, a side salad, potato, or chili. And for drinks: he can get milk, coke, sprite, or the orange drink. How many options for meals does Tanner have?
Many mp3 players can vary the order in which songs are played. Your mp3 currently only contains 8 songs (if you’re a loser). Find the number of orders in which the songs can be played.
1st Song 2nd 3rd 4th 5th 6th 7th 8th Outcomes
There are 40,320 possible song orders.
In this situation it makes more sense to use the Fundamental Counting Principle.
8The solution in this example involves the product of all the integers from n to one (n is representing the starting value). The product of all positive integers less than or equal to a number is a factorial.
• 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320
Factorial
EXAMPLE with Songs ‘eight factorial’
The product of counting numbers beginning at n and counting backward to 1 is written n! and it’s called n factorial.factorial.
8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320
FactorialSimplify each expression.
a. 4!
b. 6!
c. For the 8th grade field events there are five teams: Red, Orange, Blue, Green, and Yellow. Each team chooses a runner for lanes one through 5. Find the number of ways to arrange the runners.
4 • 3 • 2 • 1 = 24
6 • 5 • 4 • 3 • 2 • 1 = 720
= 5! = 5 • 4 • 3 • 2 • 1 = 120
5. The student council of 15 members must choose a president, a vice president, a secretary, and a treasurer.
President Vice Secretary Treasurer Outcomes
There are 32,760 permutations for choosing the class officers.
In this situation it makes more sense to use the Fundamental Counting Principle.
15 • 14 13• •12 =32,760
Let’s say the student council members’ names were: Hunter, Bethany, Justin, Madison, Kelsey, Mimi, Taylor, Grace, Maighan, Tori, Alex, Paul, Whitney, Randi, and Dalton. If Hunter, Maighan, Whitney, and Alex are elected, would the order in which they are chosen matter?
President Vice President Secretary Treasurer
Although the same individual students are listed in each example above, the listings are not the same. Each listing indicates a different student holding each office. Therefore we must conclude that the order in which they are chosen matters.
Is Hunter Maighan Whitney Alex
the same as…
Whitney Hunter Alex Maighan?
PermutationWhen deciding who goes 1st, 2nd, etc., order is important.
*Note if n = r then nPr = n!
A permutation is an arrangement or listing of objects in a specific order.
The order of the arrangement is very important!!
The notation for a permutation: nPr = n is the total number of objects r is the number of objects selected (wanted)
!
( )!
n
n r
Permutation
Notation
PermutationsSimplify each expression.
a. 12P2
b. 10P4
c. At a school science fair, ribbons are given for first, second, third, and fourth place, There are 20 exhibits in the fair. How many different arrangements of four winning exhibits are possible?
