 How is probability defined?

Activation John is at a picnic. He can choose one of three entrees, two of four vegetables and one of five desserts. How many ways can he create his meal? LOOK Familiar? How can you be sure you have all the combinations

How can we determine all the possible outcomes of a given situation?

Method One TREE DIAGRAM —an illustrative method of counting all possible outcomes. List all the choices for the 1 st event Then branch off and list all the choices for the second event for each 1 st event, etc.

 A restaurant offers a salad for $3.75. You have a choice of lettuce or spinach. You may choose one topping, mushrooms, beans or cheese. You may select either ranch or Italian dressing. How many days could you eat at the restaurant before you repeat the salad? Lettuce spinach mushrooms beans cheese mushrooms beans cheese ranch Italian ranch Italian ranch Italian ranch Italian ranch Italian ranch Italian

While the tree diagram is beneficial in that it lists every possible outcome, the more options you have the more difficult it is to draw the diagram. Fundamental counting Principle — is a mathematical version of the tree diagram, it gives the # of possible ways something can be accomplished but not a list of each way. # of ways for event 1 # of ways for event 2 # of ways for event 3...

 Example: Jani can choose from gray or blue jeans, a navy, white, green or stripped shirt and running shoes, boots or loafers? How many outfits can she wear?

Permutations —all the possible ways a group of objects can be arranged or ordered Example: There are four different books to be placed in order on a shelf. A history book (H), a math book (M), a science book (S), and an English book (E). How many ways can they be arranged? 24 WAYS = 24 H, M, S, E H, M, E, S H, S, E, M H, S, M, E H, E, M, S H, E, S, M M, E, S, H M, E, H, S M, S, H, E M, S, E, H M, H, E, S M, H, S, E S, M, E, H S, M, H, E S, H, M, E S, H, E, M S, E, M, H S, E, H, M E, M, S, H E, M, H, S E, H, M, S E, H, S, M E, S, M, H E, S, H, M

Short Cut: Factorials-- the product of all the numbers from the given number down to 1 n! = n (n-1) (n-2) (n-3) 3 (2) (1) Special Definitions 1! = 1 and 0! = 1

Sometimes we do not need to use all the objects in a given set What if you were to : EX: Arrange 3 of 4 books =24 EX: Arrange 2 of 5 books 5 4 = 20

A permutation of n objects r at a time follows the formula Example: arrange 2 of the 5 books where: n = total objects in the set r = the objects to be used

Homework worksheet 1

 What is the difference between replacement and repetition?

Activation What does replacement mean—how might that relate to probability?

Replacement—using the same object again (n r ) Example: The keypad on a safe has the digits 1- 6 on it how many: a) four digit codes can be formed _____ _____ b) four digit codes can be formed if no 2 digits can be the same _____ _____

Repetition—occurs when you have identical items in a group Example: Find all arrangements for the letters in the word TOOL TOOLOLOTLOTO TOLOOLTOLOOT TLOOOTOLLTOO OTLO OOTL OOLT We would expect 24 but since you can’t distinguish between the two O’s all possibilities with the O’s switched are removed

Formula for repetitions: where s and t represent the number of times an item is repeated EXAMPLE: How many ways can you arrange the letters in BANANAS

Circular Permutation—arranging items in a circle when no reference is made to a fixed point Example: How many ways can you arrange the numbers 1-4 on a spinner? We would expect 4! Or 24 ways but we only have 6 Circular permutations are always (n-1)! A B C D E F G? ? D

Homework worksheet 2

How can you determine the difference between a permutation and a combination?

Activation When a recipe says combine the following ingredients—what does that mean? How is that different than add the following ingredients one at a time?

Combinations—the number of groups that can be selected from a set of objects --the order in which the items in the group are selected does not matter

Example: How many three person committees can be formed from a group of 4 people—Joe, Jim, Jane, and Jill Joe, Jim, Jill Joe, Jill, Jane Joe, Jim Jane Is Joe, Jane, Jim A different committee Jim, Jane, Jill Formula: Where: n= total # of objects r = # of objects used

Homework worksheet 3

 How is an independent event defined?

Activation How are these two situations different: There are 5 marbles in a bag what is the probability of getting a red one on the second try if three are red and 2 are blue if you choose a marble and replace it? There are 5 marbles in a bag what is the probability of getting a red one on the second try if three are red and 2 are blue if you choose a marble but keep it out?

Independence means that the outcome of one event does not impact the outcome of the next event

Remember If all outcomes are successful, the probability will be 1 If no outcomes are successful, the probability will be 0 So Probability is 0 ≤ P ≤ 1

Examples: What is the probability of getting an ace from a deck of 52 cards? What is the probability of rolling a 3 on a 6 sided die?

What is the probability of rolling an even number? What is the probability of getting 2 spades when 2 cards are dealt at the same time?

What is the probability of getting a total of 5 when a red and a green die are rolled?

Homework worksheet 4

 What is meant by compound probability?

Activation What is a compound sentence in English? What about in Algebra? How might this impact probability?

OR: P(A or B) = P(A) + P(B) – P(A and B) Exclusive Events: events that do not have bearing on each other (i.e. no overlap) Example: What is the probability of getting a 2 or a 5 on the roll of a die?

Events are inclusive if they have overlap! What is the probability of drawing an ace or a heart?

AND: indicates multiplication Examples: What is the probability of tossing a three of the roll of a die and getting a head when you toss a coin? These events are independent—have no effect on the outcome of the other

Homework worksheet 5

What is a binomial expansion and how does it relate to probability?

Look at it in terms a algebra first: Expand (a + b) 3 Means =(a + b)(a + b)(a + b) =(a 2 + 2ab + b 2 )(a + b) =a 3 + 2a 2 b + ab 2 + a 2 b + 2ab 2 + b 3 =a 3 + 3a 2 b + 3ab 2 + b 3 How difficult would this be if it had been (a + b) 8 ? Activation

(a + b) 0 1 (a + b) 1 1a + 1b (a + b) 2 1a 2 + 2ab + 1b 2 (a + b) 3 1a 3 + 3a 2 b + 3ab 2 + 1b 3 The pattern for the variables is simple start with the highest power on the 1 st variable and count down, start with 0 on the second variable and count up The pattern on the coefficients is less obvious but it follows Pascal’s triangle Ones are always on the outside, but would you really want to take it to the 12 th row

Using This would make an exponent of 12 easier to do since you would only need to do the one row.

Find the 7 th term of (4x –y 2 ) 9 n = 9 r = 7-1 = (84)(4x) 5 (-y 2 ) 6 = 84 (1024x 5 )(y 12 ) = 86016x 5 y 12 n = the exponent r = the position of the term – 1 (for the 2 nd term r = 1, because we start with 0 each time) a = the part in the 1 st half of the () b = the part in the 2 nd half of the () including the sign

Find (2x + 3) 4

Find (a - 2b) 6

 How does this relate to probability? Let a = the probability that an event did not occur b = the probability that it did occur n = the # of trials r = # of success Works for any problem which is a dichotomy— something that either happens or does not happen Since total probability = 1 Then p’ = 1-p

What is the probability that a family with 9 children has 7 girls? a =.5 b =.5 n = 9 r = 7

What is the probability that a 300 hitter will hit at least 4 times in 5 hits a =.7 b =.3 n = 5 r = 4 or 5 Since total probability = 1 Then p’ = 1-p

Homework worksheet 6

Homework worksheet Review

How can we determine the probability of an event without actually running the experiment?

How what is the probability of getting 5 heads when you flip a coin ten times? Is it sufficient to flip the coin just ten times? Ten sets of ten? What if the experiment had been what is the probability of getting 5 correct answers on a ten problem true false test? Activation

 Theoretical probability  The actual probability of an event  Experimental probability  The results that you get when you run an experiment  Simulation  Used to approximate the probability when money or time is too great a factor in running an actual experiment  Design  The method used to run the simulation  Trial  One run of the method described

 A restaurant is giving away 6 actions figures to the latest movie with the purchase of each child’s meal. If you are equally like to get any figure, what is the probability of getting one of each in the purchase of ten meals? Click to roll

 A basketball player has a free throw percentage of 80%. What is the probability of making two free throws in a out of three?

How does this change our simulation?  A basketball player has a free throw percentage of 78%. What is the probability of making two free throws in a out of three?

Homework worksheet Simulations

Go to Moodle Choose Solanco High School Choose Math go to the second page, choose Mrs. Schell—Algebra III. Look for the Probability Chapter and choose the M & M project overview follow the directions