Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chance The Mean and Standard Deviation of a Probability Model • Mean of a Continuous Probability Model – Suppose the area under a density curve was cut out of solid material. The mean is the point at which the shape would balance. Law of Large Numbers As a random phenomenon is repeated a large number of times: The proportion of trials on which each outcome occurs gets closer and closer to the probability of that outcome, and The mean x¯ of the observed values gets closer and closer to μ. (This is true for trials with numerical outcomes and a finite mean μ.) Experimental Probability • Observing the results of an experiment • An event which has a 0% chance of happening (i.e. impossible) is assigned a probability of 0. • An event which has a 100% chance of happening (i.e. is certain) is assigned a probability of 1. • All other events can then be assigned a probability between 0 and 1. Experimental Probability Terminology • Number of Trials – the total number of times the experiment is repeated. • The outcomes – the different results possible for one trial of the experiment. • Frequency – the number of times that a particular outcome is observed. • Relative Frequency – the frequency of an outcome expressed as a fraction or percentage of the total number of trials. – **experimental probability = relative frequency** Sample Space • The set of all possible outcomes of an experiment • Examples: – Tossing a coin – Rolling a die Example • We roll two dice and record the up-faces in order (first die, second die) – What is the sample space S? – What is the event A: “ roll a 5”? Probability Model • Example: Rolling two dice – We roll two dice and record the up-faces in order (first die, second die) – All possible outcomes • (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) • (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) • (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) • (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) • (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) • (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) – “Roll a 5” : {(1,4) (2,3) (3,2) (4,1)} Probability Models • Give me the sample space for: Flipping two coins. Experimental Probability Examples Coin Tossing & Dice Rolling Coin Toss Dice 2-D Grids: 1. Illustrate the possible outcomes when 2 coins are tossed. 2. Illustrate the possible outcomes for the sum of 2 dice being rolled. 2-D Grid 2-D Grids: 3. Illustrate the possible outcomes when tossing a coin and rolling a die. Tree Diagrams Illustrate the possible outcomes when • tossing 2 coins • drawing 2 marbles from a bag containing red, green and yellow marbles Tree Diagrams Theoretical Probability • For fair spinners, coins or die (where a particular outcome is not weighted) the outcomes are considered to have an equal likelihood. • For a fair dice, the likelihood of rolling a 3 is the same as rolling a 5… both 1 out of 6 • This is a mathematical (or theoretical) probability and is based on what we expect to occur. – A measure of the chance of that event occurring in any trial of the experiment Warm-Up 1. Have your homework out on your desk. 2. Create a tree diagram for the following. - Flipping two coins - Pulling a marble out of a bag full of blue, green and yellow marbles. 3. How many outcomes total? 4. What is the probability of pulling a blue marble out of the bag? 5. What is the probability of flipping heads in the scenario? Homework Answers Theoretical Probability Examples • A ticket is randomly selected from a basket containing 3 green, 4 yellow and 5 blue tickets. Determine the probability of getting: – A green ticket – A green or yellow ticket – An orange ticket – A green, yellow or blue ticket Complementary Events • An ordinary 6-sided die is rolled once. Determine the chance of: – Getting a 6 – Not getting a 6 – Getting a 1 or 2 – Not getting a 1 or 2 Warm-Up 1. Have your homework out on your desk. 2. Homework Check More Grids to Find Probabilities • Use a two-dimensional grid to illustrate the sample space for tossing a coin and rolling a die simultaneously. From this grid determine the probability of: – Tossing a head – Getting a tail and a 5 – Getting tail or a 5 More Grids to Find Probabilities (cont.) • 2 circular spinners, each with 1 – 10 on their edges are twirled simultaneously. Draw a 2D grid of the possible outcomes and use your grid to determine the probability of getting – A 3 with each spinner – A 3 and a 1 – An even result for each spinner Spinner Warm-Up Compound Events • Create a 2-D grid for the following situation. • A coin is tossed and at the same time, a die is rolled. The result for the coin will be outcome A and the die, outcome B. P(A and B) P(a head and a 4) P(a head and an odd #) P(a tail and a # > 1) P(a tail and a # < 2) P(A) P(B) P(A and B) Homework Check 1. A coin is tossed three times. Find the probability that the result is at least two heads. A. 1/2 B. 1/3 B. C. 3/8 D. None of these 2. A card is drawn from a standard deck of 52 cards. Then, a second card is drawn from the deck (without replacing the first one). Find the probability that a red card is selected first and a spade is selected second. A. 1/3 B. 1/8 C. 13/102 D. None of these 3. From an urn containing 16 cubes of which 5 are red, 5 are white, and 6 are black, a cube is drawn at random. Find the probability that the cube is red or black. A. 11/16 B. 9/16 C. 15/128 D. 5/16 4. Two events that have nothing in common are called: A. inconsistent B. mutually exclusive C. complements D. Both A and B 5. A bag contains 5 white balls and 4 red balls. Two balls are selected in such a way that the first ball drawn is not replaced before the next ball is drawn. Find the probability of selecting exactly one white ball. A. 12/72 B. 20/72 C. 5/9 D. 4/5 6. A and B are two events such that p(A) = 0.2 and p(B) = 0.4. If , find . A. 0.45 B. 0.6 B. C. 0.85 D. None of these Warm-Up 1. Create a tree diagram. When you go to a restaurant you have a choice for three course meals your 4 salad choices, 6 entrees, and 5 dessert choices. 2. How many possible outcomes are there? Independent Events • Events where the occurrence of one of the events does not affect the occurrence of the other event. • In general, if A and B are independent events, then – P(A and B) = P(A) x P(B) • Ex: a coin and a die are tossed simultaneously. Determine the probability of getting a head and a 3 without using a grid. Using Tree Diagrams Examples: Examples (cont.): • Carson is not having much luck lately. His car will only start 80% of the time and his moped will only start 60% of the time. – Draw a tree diagram to illustrate the situation. • 1st set of branches for the car, 2nd set of branches for the moped – Use the diagram to determine the chance that • Both will start • He has to take his car. • He has to take the bus. Dependent Events • Think About It: A hat contains 5 red and 3 blue tickets. One ticket is randomly chosen and thrown out. A second ticket is randomly selected. What is the chance that it is red? • Not independent; the occurrence of one of the events affects the occurrence of the other event. • If A and B are dependent events then P(A then B) = P(A) x P(B given that A has occurred) Examples: • A box contains 4 red and 2 yellow tickets. Two tickets are randomly selected one by one from the box, without replacement. Find the probability that: – Both are red – The first is red and the second is yellow Examples (cont.): • A hat contains tickets with numbers 1 – 20 printed on them. If 3 tickets were drawn from the hat without replacement, determine the probability that all are prime numbers. Examples (cont.): • A box contains 3 red, 2 blue and 1 yellow marble. Draw a tree diagram to represent drawing 2 marbles. • With replacement Without replacement • Find the probability of getting two different colors: – If replacement occurs – If replacement does not occur Examples (cont.): • A bag contains 5 red and 3 blue marbles. Two marbles are drawn simultaneously from the bad. Determine the probability that at least one is red. Z Is it a fair game? Questions will be put up on the board For each, you have to decide if the game is: Fair Not Fair Once you decide on your answer, write it on your mini-whiteboard Only show the you answer when asked Z Is it a fair game? Three people have boards like the one shown below. You throw a coin onto a board, if it lands on a shaded square you win (assume the coin lands exactly in a square) Z Is it a fair game? A marble is picked from the container by the teacher If its red the girls get a point, if its blue the boys get a point Z Is it a fair game? Nine cards numbers 1 to 9 are used for a game 1 2 3 4 5 6 7 8 9 A card is drawn at random If a multiple of 3 is drawn team A gets a point If a square number is drawn team B gets a point If any other number is drawn team C gets a point Z Is it a fair game? A spinner has 5 equal sectors numbers 1 to 5, it is spun many times 5 If the spinner stops on an even number 1 team A gets 3 points 4 If the spinner stops on an odd number team B gets 2 points 2 3 Warm Up 1. Get your homework out. 2. A box contains 4 red marbles, 5 blue marbles and 1 green marble. We select 2 marbles without replacement. Determine the probability of getting: – At least 1 red marble – One green and one blue marble Sets & Venn Diagrams • A Venn diagram consists of a rectangle which represents the sample space and at least 1 circle within it representing particular events. Examples • The Venn diagram represents a sample space of students. The event E, shows all those that have blue eyes. Determine the probability that a student – Has blue eyes Examples (cont.) • Draw a Venn diagram and shade the regions to represent the following: – 1. In A but not in B 2. Neither in A nor B • A B denotes the union of the sets A and B. – A or B or both A and B. • A B denotes the intersection of sets A and B. – All elements common to both sets. • Disjoint sets do not have elements in common. So A B , where represents the empty set. – A and B are said to be mutually exclusive. Examples (cont.) • If A is the set of all factors of 36 and B is the set of all factors of 54, find: –AUB –A∩B Examples (cont.) Examples (cont.) • In a class of 30 students, 19 study Physics, 17 study Chemistry and 15 study both. Display this in a Venn diagram and find the probability that a student studies: – Both – At least 1 of the subjects – Physics, but not Chemistry – Exactly one of the subjects – Neither – Chemistry given that the student also studies physics Warm-Up Find the following probabilities. Laws of Probability • For 2 events A and B, – P (A U B) = P(A) + P(B) – P (A ∩ B) • Example: P(A) = 0.6, P(A U B) = 0.7 and P(A ∩ B) = 0.3 – Represent this using a Venn diagram and find P(B) Mutually Exclusive Events • If A and B are mutually exclusive the intersection is the empty set and equals 0. – So the law becomes: • Example: A box of chocolate contains 6 with hard centers (H) and 12 with soft centers (S). – Are H and S mutually exclusive? – Find P(H ∩ S) – Find P(H U S) Laws of Probability (cont.) • Conditional Probability (dependent events): – A | B represents “A occurs knowing B has occurred” – It follows that: Example • In a class of 40, 34 like bananas, 22 like pineapples, and 2 dislike both fruits. Find the probability that a student: – Likes both – Likes at least one – Likes bananas give that they like pineapples – Dislikes pineapples given that they like bananas Example (cont.) • Box A contains 3 red and 2 white tickets. Box B contains 4 red and 1 white. A die with 4 faces marked A and 2 faces marked B is rolled and used to select a box. Then we draw a ticket. Find the probability that: – The ticket is red – The ticket was chosen from B given it is red. Using Definitions • If A and B are independent, how do we find P(A and B)? – When 2 coins are tossed, A is the event of getting 2 heads. When a die is rolled, B is the event of getting a 5 or 6. Prove that A & B are independent Using Definitions (cont.) • If A and B are mutually exclusive, what has to be true? – P(A) = ½ and P(B) = 1/3, find P(A U B) if: • A and B are mutually exclusive • A and B are independent