Probability and Chance Random Experiment l An experiment

Probability and Chance

Random Experiment l An experiment is random if – The outcome depends on chance (we are not sure of the outcome (result)) – We can list all the possible outcomes in the sample space Example: Roll a die. We are not sure what number will be face up but we know it will be one of = {1 , 2 , 3 , 4 , 5, 6}

Event l An event is a subset of the sample space l Ex : Event A is an even number – A { 2, 4, 6} Ex: Event B is a number less than 5 - B { 1, 2, 3, 4}

Set Theory A ∩ B is A intersect B A ∪B is A union B

Compatible l Compatible events can occur at the same time l Incompatible events can not occur at the same time A and B can have a shared outcome l A∩B ( A intersect B) is {2, 4} l Event C is roll a 5 the C in incompatible with both A and B l A ∩ C is empty B ∩ C is empty l

Complementary Event l Events A and B are complementary if together they make up the sample space and there is nothing shared l. AUB= l. A ∩ B=0 l Example: A roll an even number l B roll an odd number

Probability (P) l Probability is a measure of how likely it is for an event to happen. l We name a probability with a number from 0 to 1. • If an event is certain to happen, then the probability of the event is 1. P=1 • If an event is certain not to happen, then the probability of the event is 0. P=0

Probability l If it is uncertain whether or not an event will happen, then its probability is some fraction between 0 and 1 (part ÷ whole). l Part = # of possible favorable outcomes l Whole = # of all possible outcomes

B C 3 A D 2 1 A C B 1. What is the probability that the spinner will stop on part A? 2. What is the probability that the spinner will stop on (a) An even number? (b) An odd number? 3. What fraction names the probability that the spinner will stop in the area marked A?

Probability Questions l Lawrence is the captain of his track team. The team is deciding on a color and all eight members wrote their choice down on equal size cards. If Lawrence picks one card at random, what is the probability that he will pick blue? blue yellow red green black blue black

Donald is rolling a number cube labeled 1 to 6. Which of the following is LEAST LIKELY? A. an even number B. an odd number C. a number greater than 5 l

CHANCE what are the odds? l Chance is how likely it is that something will happen. To state a chance, we use a percent or a ratio ( part : part) 0 ½ 1 Probability Certain not to happen Equally likely to happen or not to happen Certain to happen Chance 0% 50: 50 100%

Chance l When a meteorologist states that the chance of rain is 50%, the meteorologist is saying that it is equally likely to rain or not to rain. If the chance of rain rises to 80%, it is more likely to rain. If the chance drops to 20%, then it may rain, but it probably will not rain.

1 2 4 3 1. What is the chance of spinning a number greater than 1? 4 1 2 3 5 2. What is the chance of spinning a 4? 3. What is the chance that the spinner will stop on an odd number? 4. What is the chance of rolling an even number with one toss of on number cube?

Sample Spaces A sample set refers to the complete set of all the possible outcomes Example: Roll a die. What are all the possible outcomes? Sample set “S” S = {1, 2, 3, 4, 5, 6}

Sample Spaces Example: Toss a coin. What are all the possible outcomes? Sample space “S” S = {H, T} Toss two coins. What is the sample space? S = {HH, HT, TH, TT}

Events A set of outcomes is referred to as an event. A specific outcome (part of the whole) For example, when rolling a die the outcomes that are an even number would be referred to as an event. Event = {2, 4, 6} S = {1, 2, 3, 4, 5, 6} It is clear that outcomes and events are subsets of the sample space, S.

Sample Space versus Events We use the symbol omega instead of S so that we don’t get mixed up with events Events are given a capital letter Ex = {1, 2, 3, 4, 5, 6} A = { 2, 4, 6} The sample space is all the possible outcomes of rolling a dice The event A is rolling an even number.

Compound Events Sometimes we are asked to find the probability of one event OR another Sometimes we are asked to find the probability of one event AND another What’s the difference? Example: What is the probability of rolling a 2 OR a 4?

Compound Problems: Multiple Events What is the probability of rolling a 2 and a 4 if two die are rolled? S = {11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66} Event {2 and 4} = {24, 42} All possible outcomes = 36 Possible outcomes of the stated event = 2 Therefore the probability is 2 out of 36 P = 0. 056

Compound Events S = {1, 2, 3, 4, 5, 6} Event {2} or {4} There are 2 possible outcomes out of 6 P = 2/6 P = 0. 33

Logical connectors And , Or When we see the probability of event A and B we multiply When we see the probability of event A or B, we add Example: We roll a die, what is the probability of rolling a 3 or a 5? 1/6 + 1/6 = 2/6 or 0. 33 Example: We roll a die and then roll it again, what is the probability of rolling a 3 and a 5? 1/6 x 1/6 = 1/36 (much less likely)

Compound Events Independent versus Dependent Events Independent: if event A does not influence the probability of event B Dependent: if event A does influence the probability of event B Example: Event A: choose a marble Event B: choose a marble They are independent if I replace the marble, dependent if I do not replace the marble

Compound Events Independent versus Dependent Events Example: there are 100 skittles 20 red 20 orange 20 green 20 purple 20 yellow What is the probability of choosing a red one, eating it and then choosing a yellow one? Are these events dependent or independent?

Compound Events Independent versus Dependent Events Example: there are 100 skittles 20 red 20 orange 20 green 20 purple 20 yellow What is the probability of choosing a red one, eating it and then choosing a yellow one? P(A) X P(B) = 20/100 X 20/99 (remember, I ate one)

Compound Events Independent versus Dependent Events Example: there are 100 skittles 20 red 20 orange 20 green 20 purple 20 yellow What is the probability of choosing 2 red one (I don’t replace the first – obviously) P(A) X P(B) = 20/100 X 19/99 (remember, I ate one)

Compound Events Independent versus Dependent Events Example: there are 100 skittles 20 red 20 orange 20 green 20 purple 20 yellow What is the probability of eating 1 orange, 1 green, 1 purple and then 1 green?

20/100 x 20/99 x 20/98 x 19/97 Get it?

Compound Events Independent versus Dependent Events Example: there are 100 skittles 20 red 20 orange 20 green 20 purple 20 yellow Create your own question…