Probability Simple and Compound Probability Counting: Permutations and Combinations

Vocabulary Probability: The likelihood or chance of something happening Outcome: a possible result of an experiment Event: an outcome or a collection of outcomes Sample Space: the set of all possible outcomes

Probability Probability can have two approaches Experimental Theoretical

Experimental Probability Experimental probability is found by repeating an experiment and observing the outcomes. Experimental probability: P(event) = number of times event occurs (successes) total number of trials

Theoretical Probability All outcomes are equally likely to occur. Theoretical probability: P(E) = number of favorable outcome total number of possible outcomes

How can you tell which is experimental and which is theoretical probability? You tossed a coin 10 times and recorded a head 3 times, a tail 7 times P(head)= 3/10 P(tail) = 7/10 Theoretical: Toss a coin and getting a head or a tail is 1/2. P(head) = 1/2 P(tail) = 1/2

Identifying the Type of Probability A bag contains three red marbles and four blue marbles. P(red) = 3/7 Theoretical (The result is based on the possible outcomes)

Identifying the Type of Probability You draw a marble out of the bag, record the color, and replace the marble. After 6 draws, you record 2 red marbles P(red)= 2/6 = 1/3 Experimental (The result is found by repeating an experiment.)

Examples You spin a spinner 30 times. It stops on 12 three times. What is the probability of stopping on a 12? What is the probability of drawing a King from a deck of cards? You roll a 6-sided number cube, what is the probability of landing on an even number?

Odds: Odds of an event compare the number of favorable and unfavorable outcomes Odds in favor: Number of favorable outcomes Number of unfavorable outcomes Odds against: Number of unfavorable outcomes Number of favorable outcomes

Example A spinner has numbers 1-6. What are the odds in favor of stopping on a multiple of 2? A spinner has numbers 1-6. What are the odds against stopping on a number less than 2?

Compound Events Combines two or more events, using the word and or the word or. Mutually Exclusive Events: no common outcomes (or) Overlapping Events: at least one common outcome (will need to subtract the common outcome)

Visual Mutually Exclusive P(A)+P(B) Overlapping Events P(A)+P(B)-P(A and B)

Examples You roll a six sided number cube. Find the probability that you roll a 2 or an odd number. You roll a number cube. Find the probability that you roll an even number or a prime number.

Examples What is the probability I draw a King or an 8 from a deck of cards? What is the probability I draw a King and a spade from a deck of cards?

Independent and Dependent Events Independent Events: two events are independent if the occurrence of one event has no effect on the other event (replacement) Dependent: two events are dependent if the occurrence of one event affects the occurrence of the other (without replacement)

Example A bag of marbles has 2 red, 4 green, and 1 blue. Find the probability of picking a green marble, replacing it, and then picking a blue marble. A bag of marbles has 2 red, 4 green, and 1 blue. Find the probability of picking a green marble, not replacing it, and then picking a blue marble.

Examples I draw two cards from a deck of cards. What is the probability that both cards are Kings? I draw a card from a deck of cards and then replace it and draw another card. What is the probability both are Kings?

Fundamental Counting Principle Suppose most of your clothes are dirty and you are left with 2 pants and 3 shirts. How many choices do you have or how many different ways can you dress? Total number of ways different events can occur

Examples An ice cream store sells 5 varieties of ice cream. The shop has three different cones and 4 different toppings. How many different ice cream cones can you make?

Permutations and Combinations Permutation: an arrangement of objects in which order is important Combination: a selection of objects where order is not important

Difference: List possibilities of ordering two letters from A, B, C, D Permutations: order matters!!! AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC 12 outcomes Combinations: order doesn’t matter!! Take out repeats! AB, AC, AD, BC, BD, CD 6 outcomes

Permutation Example There are 4 runners in a race. How many different ways can they finish the race?

Combination Example Maria has 5 different shirts---a green shirt, a pink shirt, a yellow shirt, a white shirt, and a blue shirt. She needs to pack three of them for a trip, and she doesn’t care which three she takes. How many different combinations of the three shirts can she pack?

Examples: Identify as a combination or permutation first How many ways can you arrange the letters in the word JULY? How many ways can you arrange 2 of the letters in the word JULY?

Examples: Identify as a combination or permutation first Your band has written 12 songs and plans to record 9 of them for a CD. In how many ways can you arrange the songs on the CD?

Examples: Identify as a combination or permutation first Four students from your class of 120 students will be selected to organize a fundraiser. How many groups of 4 students are possible?

Examples: Identify as a combination or permutation first In how many ways can 5 people sit in a car that holds 5 passengers?

Examples: Identify as a combination or permutation first Ten students are auditioning for 3 different roles in a play. In how many ways can the 3 roles be filled?

Examples: Identify as a combination or permutation first You order a sandwich at a restaurant. You can choose 2 sides dishes from a list of 8. How many combinations of side dishes are possible?