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Probability and Statistics. Chapter 12. Multiplication Counting Principle. (The Fundamental Counting Principle) Multiply the number of choices for each Example: You have a five shirts and three pairs of pants. How many outfits do you have?. Pant choices. Shirt choices. Examples:.

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multiplication counting principle
Multiplication Counting Principle

(The Fundamental Counting Principle)

Multiply the number of choices for each

Example: You have a five shirts and three pairs of pants. How many outfits do you have?

Pant choices

Shirt choices

examples
Examples:
  • A restaurant has 4 appetizers, 8 main courses, and 3 desserts. How many different dinners are available?
  • A pizza shop offers 8 vegetable toppings and 6 meat toppings. How many different pizzas can you make with one of each topping?
factorial
Factorial

n!

Multiply the values 1 – n

Example: 4!

= 4 x 3 x 2 x 1

= 24

examples find the value
Examples: Find the value
  • 5!
  • 7!
  • Find 12! using the calculator

In the Calculator: Go to MATH, PRB, and choose 4

permutation
Permutation
  • An arrangement of objects in a specific order

Example: If you are arranging the letters A, B, and C without repeating the letters, how many ways can they be arranged?

ABC, ACB, BAC, BCA, CAB, CBA

ORDER MATTERS!!!!

permutation notation
Permutation notation:

nPr

Where n is the number of objects and r is how many objects are being arranged

example
Example
  • You have 8 books and are arranging 3 of them on a shelf. How many ways can you arrange the three books.

n = the number of objects

n = 8

r = how many are being arranged

r = 3

example1
Example
  • There are 12 different songs that the Spring Chorus can choose from. How many ways can they perform 4 of the songs?
example2
Example
  • There are 6 boys in the class. Three of the boys are going to the board to do homework problems. How many ways can the boys be called to the board?
combination
Combination
  • An arrangement of objects in where order does not matter

Example: If you are arranging the letters A, B, and C without repeating the letters, how many ways can they be arranged?

ABC, ACB, BAC, BCA, CAB, CBA are all the same!! So there is only 1 way!

ORDER Doesn’t MATTER!!!!

combination notation
Combination notation:

nCr

Where n is the number of objects and r is how many objects are being arranged

example3
Example
  • You have 8 books and are arranging 3 of them on a shelf. How many ways can you arrange the three books where order does not matter?

n = the number of objects

n = 8

r = how many are being arranged

r = 3

calculating npr1
Calculating nPr

Removes the repeats!

example4
Example
  • There are 12 different songs that the Spring Chorus can choose from. How many combinations of 4 songs can they perform?
example5
Example
  • There are 6 boys in the class. Three of the boys are going to the board to complete the same problem. How many ways can the boys be called to the board considering they are all doing the same problem?
probability
Probability
  • How likely it is that an event with occur

Notation: P(event)

Example: What is the probability that you will roll a 5 on a fair dice?

Outcome – the result of a single trial

5

Sample Space – all possible outcomes

1, 2, 3, 4, 5, 6

Event – the outcome

P(5) =

slide19

Probability can be represented as a fraction, or decimal and is always between 0 and 1.

P(5) = = .167

certain

Equally likely

impossible

Less likely

More likely

How likely is it that a 5 will be rolled?

Less likely

theoretical probability
Theoretical Probability
  • Probability based on equally likely outcomes
  • Example: Rolling the dice for a 5 provides a theoretical probability of
experimental probability
Experimental Probability
  • Probability based on data collected
  • Example: Rolling the dice six different times. How many times did you roll a 5?
example6
Example:
  • Find the theoretical probability of flipping tails on a coin.
  • Flip the coin ten times. What is the experimental probability?
  • How do they compare?
complement of an event
Complement of an event
  • The probability of all the outcomes that are not in the event

P(event) + P(not event) = 1

P(5) + P(not 5) = 1

in a deck of cards what is the probability that the you will be dealt a card that is not a heart
In a deck of cards, what is the probability that the you will be dealt a card that is not a heart?

P(heart)

P(not heart)

in a deck of cards what is the probability that the you will be dealt a card that is not a queen
In a deck of cards, what is the probability that the you will be dealt a card that is not a queen?

P(queen)

P(not queen)

finding the odds

Leave as a fraction

Finding the odds

Ratio that compares the favorable vs. unfavorable outcomes.

Odds in favor of an event =

Odds against an event =

what are the odds that you will roll a 5 on a dice
What are the odds that you will roll a 5 on a dice?

Favorable Events = 5

# Favorable Events = 1

Unfavorable Events = 1, 2, 3, 4, 6

# Unfavorable Events = 5

what are the odds that you will not roll a 5 on a dice
What are the odds that you will not roll a 5 on a dice?

Favorable Events = 1, 2, 3, 4, 6

# Favorable Events = 5

Unfavorable Events = 5

# Unfavorable Events = 1