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1. 45, 46, 56 2.Combination – order is not important 3.Permutation – order is important 4. 5.ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE 6.ABCD, ABCE, ABCF, ABDE, ABDF, ABEF, ACDE, ACDF, ACEF, ADEF, BCDE, BCDF, BCEF, BDEF, CDEF 7.13, 15, , 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, , 12346, 12356, 12456, 13456, ABCD COMBINATI ON 1XX AB 2X X AC 3X XAD 4 XX BC 5 X XBD 6 XXCD

RS, RT, ST JK, JL, JM, KL, KM, LM JKL, JKM, JLM, KLM Combinations – Permutations – Permutations Combinations A C a 1/6 b 1/2

Permutations with Repetitions Compound Events

Permutations with Repetitions Permutation Formula The number of permutations of “n” objects, “r” of which are alike, “s” of which are alike, ‘t” of which are alike, and so on, is given by the expression

Permutations with Repetitions Example 1: In how many ways can all of the letters in the word SASKATOON be arranged? Solution: If all 9 letters were different, we could arrange then in 9! Ways, but because there are 2 identical S’s, 2 identical A’s, and 2 identical O’s, we can arrange the letters in: Therefore, there are different ways the letters can be arranged.

Permutations with Repetitions Example 2: Along how many different routes can one walk a total of 9 blocks by going 4 blocks north and 5 blocks east? Solution: If you record the letter of the direction in which you walk, then one possible path would be represented by the arrangement NNEEENENE. The question then becomes one to determine the number of arrangements of 9 letters, 4 are N’s and 5 are E’s.  Therefore, there are 126 different routes.

Probability of Compound Events

Vocabulary:  Replacement – after picking a card or selecting something, it is put back and the next time something is selected, there are the same number of possibilities. Pick an Ace–(4/52) replace it and pick a king(4/52) Pick an Ace (4/52) replace it and pick another ace (4/52)  Without Replacement – the item selected is kept and not replaced so there is one less item to select. Pick an Ace–(4/52) DON’T replace it and pick a king(4/51) Pick an Ace (4/52) DON’T replace it and pick another ace (3/51)

What are COMPOUND EVENTS? - There are (2) types of compound events: (1) Independent Events – involves two or more events in which the outcome of one event DOES NOT affect the outcome of any other events Examples: roll dice, coin flip, problems with replacement P(A and B) = P(A) x P(B) Probability of Compound Events

What are COMPOUND EVENTS? (2) Dependent Events - involves two or more events in which the outcome of one event DOES affect the outcome of any other events Examples: deck of cards, selecting item from container, problems without replacement P(A and B) = P(A) x P(B following A) Probability of Compound Events

Mr. Kim needs two students to help him with a science demonstration for his class of 18 girls and 12 boys. He randomly chooses one student who comes to the front of the room. He then chooses a second student from those still seated. What is the probability that both students chosen are girls?

Probability of Compound Events A compound event consists of two or more simple events. Examples: rolling a die and tossing a penny spinning a spinner and drawing a card tossing two dice tossing two coins

Compound Events When the outcome of one event does not affect the outcome of a second event, these are called independent events. The probability of two independent events is found by multiplying the probability of the first event by the probability of the second event.

Compound Event Notations

Compound Probability 1)P(roll even #, spin odd) =

Probability of Compound events P(jack, tails)

Compound Events Events that cannot occur at the same time are called mutually exclusive. Suppose you want to find the probability of rolling a 2 or a 4 on a die. P(2 or 4) Since a die cannot show both a 2 and a 4 at the same time, the events are mutually exclusive.

Compound Mutually Exclusive

Mutually Exclusive Example: Alfred is going to the Lakeshore Animal Shelter to pick a new pet. Today, the shelter has 8 dogs, 7 cats, and 5 rabbits available for adoption. If Alfred randomly picks an animal to adopt, what is the probability that the animal would be a cat or a dog?

Since a pet cannot be both a dog and a cat, the events are mutually exclusive. Add. The probability of randomly picking a cat or a dog is

Example Two: The French Club has 16 seniors, 12 juniors, 15 sophomores, and 21 freshmen as members. What is the probability that a member chosen at random is a junior or a senior?

Compound Probability Mutually inclusive When two events, A and B, are non-mutually exclusive, the probability that A or B will occur is: P(A or B) = P(A) + P(B) - P(A and B) The P(king or diamond) (there is a king of diamonds that can only be counted once) This is called mutually inclusive.

Independent Practice: Questions. (1) Wyatt has four $1 bills in his wallet and three $10 bills in his wallet. What is the probability he will reach into his wallet twice and pull out a $1 bill each time? (Assume he does not replace the first bill) (2) A bag contains 3 green and 2 purple marbles. What is the probability of drawing two purple marbles in a row from the bag if the first marble is replaced? Probability of Compound Events

Independent Practice: Answers. (1)$1 $1 $1 $1$10 $10 $ (2) P(purple,then purple) = 2/5 x 2/5 = 4/25 Probability of Compound Events P($1,then $1) = x = =

Summary: The difference between simple and compound events: (1) simple event – a specific outcome or type of outcome. (2) compound event – events which consist of two or more simple events. Probability of Compound Events

Summary: The difference between independent and dependent events: (1) independent event – two or more simple events in which the outcome of one event DOES NOT affect the outcome of other event(s) (2) dependent event – two or more simple events in which the outcome of one event DOES affect the outcome of other event(s) Probability of Compound Events

Real World Example: Joanna had 3 roses, 4 tulips, and 1 carnation in a vase. She randomly selected one flower, took a photo of it, and put it back. She then repeated the steps. What is the probability that she selected a rose both times? Probability of Compound Events P(rose,then rose) = x =

Homework: worksheet Probability of Compound Events