You probability wonder what we’re going to do next!

Probability Basics Experiment Experiment An activity with observable results or outcomes An activity with observable results or outcomes Sample space Sample space The set of all possible outcomes for an experiment The set of all possible outcomes for an experiment Event Event Any subset of the sample space Any subset of the sample space

Probability Basics — General Definition where P(E) represents the probability of an event E occurring, n(E) represents the number of individual outcomes in the event E, and n(S) represents the number of individual outcomes in the sample space S.

Flip a coin A well-known statistician named Karl Pearson once flipped a coin 24,000 times and recorded ________ “heads”; this result is extremely close to the theoretical expected value! A well-known statistician named Karl Pearson once flipped a coin 24,000 times and recorded ________ “heads”; this result is extremely close to the theoretical expected value! P(H) = _____ P(T) = _____ P(H) = _____ P(T) = _____ Expected # of H = P(H) x 24,000 = _____ Expected # of H = P(H) x 24,000 = _____

Spinners Spin each spinner once. Find the probability that the spinner lands in region A. A B C A A B CD

Spinners If S = {1, 2, 3, 4, 5,..., 22, 23, 24}, find the probability of the spinning of a Prime number Prime number Even number Even number Number less than 10 Number less than 10 Number less than 3 or greater than 17 Number less than 3 or greater than 17 Number less than 12 and greater than 9 Number less than 12 and greater than 9

Rolling Dice Roll a single die once. Find the following probabilities: P(number greater than 4 or less than 2) P(number greater than 4 or less than 2) P(odd or even number) P(odd or even number) P(number greater than 10) P(number greater than 10) P(at least 3) P(at least 3)

Probability Vocabulary Complementary event Complementary event Everything else (besides the outcomes in the event) in the sample space Everything else (besides the outcomes in the event) in the sample space Examples: Examples: If A = “roll a 1 or a 2 on a die”, then “A complement” = “roll a 3, 4, 5, or 6 on a die”. If A = “roll a 1 or a 2 on a die”, then “A complement” = “roll a 3, 4, 5, or 6 on a die”. If R = “it rains today”, then R complement = “it doesn’t rain today”. If R = “it rains today”, then R complement = “it doesn’t rain today”.

Standard Cards Find the probability of drawing Find the probability of drawing an ace from a standard deck of playing cards. Find P(“face card”) Find P(“face card”) Find P(card with a value between 4 and 9) Find P(card with a value between 4 and 9)

More Vocabulary Mutually exclusive events (Disjoint sets): Mutually exclusive events (Disjoint sets): When one event occurs, the other cannot possibly occur; the events have no overlap When one event occurs, the other cannot possibly occur; the events have no overlap Example: Example: If A = “roll an even number” and B = “roll a 3 or a 5”, find P(A or B) and find P(A and B). If A = “roll an even number” and B = “roll a 3 or a 5”, find P(A or B) and find P(A and B).

Probability of A or B Mutually exclusive events Mutually exclusive events Non-mutually exclusive events Non-mutually exclusive events

Probability of A or B Draw a card out of a standard 52-card deck. Find the probability that the card is either:(a) a black card or an ace (b) a red card or a club Draw a card out of a standard 52-card deck. Find the probability that the card is either:(a) a black card or an ace (b) a red card or a club Roll a die once. If A = “roll an even number” and B = “roll a 5 or a 6”, find P(A or B). Roll a die once. If A = “roll an even number” and B = “roll a 5 or a 6”, find P(A or B).

Fundamental Counting Principle If event M can occur in m ways and after it has occurred, event N can occur in n ways, then event M followed by event N can occur in m x n ways. If event M can occur in m ways and after it has occurred, event N can occur in n ways, then event M followed by event N can occur in m x n ways. (P.S. A tree diagram helps!) (P.S. A tree diagram helps!)

Fundamental Counting Principle How many outcomes are there for flipping 3 coins? How many outcomes are there for flipping 3 coins? How many outcomes are there for rolling 2 dice? How many outcomes are there for rolling 2 dice? If I have 6 pairs of pants and 8 shirts from which to choose, how many outfits can I pick? If I have 6 pairs of pants and 8 shirts from which to choose, how many outfits can I pick?

Fundamental Counting Principle If automobile license plates consist of 4 letters followed by 3 digits (and repetition of letters and digits is allowed), how many different license plates are possible? If automobile license plates consist of 4 letters followed by 3 digits (and repetition of letters and digits is allowed), how many different license plates are possible? (This time, a tree diagram isn’t encouraged.)

Multi-stage Experiments For any multi-stage experiment, the probability of the outcome along any path of the tree diagram is equal to the product of the probabilities along the path. For any multi-stage experiment, the probability of the outcome along any path of the tree diagram is equal to the product of the probabilities along the path.

Toss 2 coins List the sample space. Use set notation and a tree diagram. List the sample space. Use set notation and a tree diagram. Find the probability of Find the probability of at least one head.

The Problem If the chance for success on the first stage of a rocket firing procedure is 96%, the chance for success on the second stage is 98%, and the chance for success on the final stage is 99%, find the probability for success on all 3 stages of the rocket firing procedure. If the chance for success on the first stage of a rocket firing procedure is 96%, the chance for success on the second stage is 98%, and the chance for success on the final stage is 99%, find the probability for success on all 3 stages of the rocket firing procedure.

Rolling Two Dice: Sample Space

Rolling Two Dice Find the probability of a 3 on the first roll and a 3 on the second roll of a die. Find the probability of a 3 on the first roll and a 3 on the second roll of a die. Find the probability of a sum of 7. Find the probability of a sum of 7. Find the probability of a sum of 10 or more. Find the probability of a sum of 10 or more. Find the probability that both numbers are even. Find the probability that both numbers are even.

Independent Events When the outcome of one event has no influence on the outcome of a second event, the events are independent. When the outcome of one event has no influence on the outcome of a second event, the events are independent. For any independent events A and B, For any independent events A and B, P(A and B) = P(A) x P(B).

Draw a ball from a container, replace it, and then draw a 2 nd ball. Find the probability of a red, then a red. Find the probability of a red, then a red. Find P(no ball is red). Find P(no ball is red). Find P(at least one red). Find P(at least one red). Find P(same color). Find P(same color).

Draw a ball from a container, don’t replace it, and then draw a 2 nd ball. (dependent events) Find P(red, then green). Find P(red, then green). Find P(no ball is red). Find P(no ball is red). Find P(same color ball). Find P(same color ball).

A bag contains the letters of the word “probability”. Draw 4 letters, one by one, from the bag. Find the probability of picking the letters of the word “baby” if the letters are drawn Draw 4 letters, one by one, from the bag. Find the probability of picking the letters of the word “baby” if the letters are drawn With replacement With replacement Without replacement Without replacement

Geometric Probabilities If a dart hits the target below, find the probability that it hits somewhere in region 1. If a dart hits the target below, find the probability that it hits somewhere in region The radius of the inner circle is 1 unit, and the radius of the outer circle is 2 units.

For a challenge, or two, or three! “Pascal’s Probabilities” “Pascal’s Probabilities” “The Prisoner Problem” “The Prisoner Problem” “The Birthday Problem” “The Birthday Problem”

Using Simulations Flipping a coin Flipping a coin Rolling a die Rolling a die Find the probability of a married couple having 2 boys and 2 girls. Find the probability of a married couple having 2 boys and 2 girls.

Isn’t that odd?

Odds Find the odds for tossing a “head” on a fair coin. Find the odds for tossing a “head” on a fair coin. Find the odds for rolling a sum of 7 on the roll of two dice. Find the odds for rolling a sum of 7 on the roll of two dice. Find the odds for drawing a card valued from 1 (ace) to 8, inclusive, from a standard 52-card deck. Find the odds for drawing a card valued from 1 (ace) to 8, inclusive, from a standard 52-card deck.

Conditional Probabilities When the sample space of an experiment is affected by additional information When the sample space of an experiment is affected by additional information

Conditional Probabilities If A = “getting a tail on the 1 st toss of a coin” and B = “getting a tail on all three tosses of a coin”, find P(B|A). If A = “getting a tail on the 1 st toss of a coin” and B = “getting a tail on all three tosses of a coin”, find P(B|A). What is the probability of rolling a 6 on a fair die if you know that you rolled an even number? What is the probability of rolling a 6 on a fair die if you know that you rolled an even number?

Expected Value If, in an experiment, the possible outcomes are numbers a 1, a 2, a 3,..., a n occurring with probabilities p 1, p 2, p 3,..., p n, respectively, then the expected value, E, is given by the equation If, in an experiment, the possible outcomes are numbers a 1, a 2, a 3,..., a n occurring with probabilities p 1, p 2, p 3,..., p n, respectively, then the expected value, E, is given by the equation E = a 1 p 1 + a 2 p 2 + a 3 p , + a n p n. E = a 1 p 1 + a 2 p 2 + a 3 p , + a n p n.

Expected Value (level 1) Flip a coin 1,000 times. How many heads do you expect? Flip a coin 1,000 times. How many heads do you expect? Roll a pair of dice 60 times. How many times do you expect a sum of 5? Roll a pair of dice 60 times. How many times do you expect a sum of 5?

Expected Value (level 2) If a player gets $2 if the spinner lands on A, $4 for landing on B, $4 for C, and If a player gets $2 if the spinner lands on A, $4 for landing on B, $4 for C, and $1 for D, what is the expected payoff for this game? If the game costs $3 to play, is this a fair game? If the game costs $3 to play, is this a fair game? A A B CD

Factorial Notation 0! = 1 (by definition) 0! = 1 (by definition) Compute: Compute:

Permutations From n objects, choose r of them and arrange them in a definite order. The number of ways this can be done is given by From n objects, choose r of them and arrange them in a definite order. The number of ways this can be done is given by

Permutations (Correspondences) How many different How many different ways can 4 swimmers (Al, Betty, Carol, and Dan) be placed in 4 lanes for a swim meet?

Permutations If there are 12 players on a little league baseball team, how many ways can the coach arrange batting orders, with 9 positions in the field and at bat? If there are 12 players on a little league baseball team, how many ways can the coach arrange batting orders, with 9 positions in the field and at bat?

Combinations From n objects, choose subsets of size r (order is unimportant). The number of ways this can be done is given by From n objects, choose subsets of size r (order is unimportant). The number of ways this can be done is given by

Combinations With 9 club members, how many different committees of 4 can be selected to attend a conference? With 9 club members, how many different committees of 4 can be selected to attend a conference? Braille Activity Braille Activity

Permutations & Combinations How many games are played in a women’s soccer conference if there are 8 teams and all teams play one another once? How many games are played in a women’s soccer conference if there are 8 teams and all teams play one another once? There are 10 members of a club. How many different “slates” could the membership elect as president, vice- president, and secretary/treasurer (3 offices)? There are 10 members of a club. How many different “slates” could the membership elect as president, vice- president, and secretary/treasurer (3 offices)?

Probability (with permutations & combinations) Given a class of 12 girls and 9 boys, Given a class of 12 girls and 9 boys, In how many ways can a committee of 5 be chosen? In how many ways can a committee of 5 be chosen? In how many ways can a committee of 3 girls and 2 boys be chosen? In how many ways can a committee of 3 girls and 2 boys be chosen? What is the probability that a committee of 5, chosen at random, consists of 3 girls and 2 boys? What is the probability that a committee of 5, chosen at random, consists of 3 girls and 2 boys?