Transcript Slide 1
Pick Me Basic Probability Population – a set of entities concerning which statistical inferences are to be drawn Random Sample – all member of population and all group members of a given size have equal chance of being in the sample Sample Space- possible outcomes of an experiment Basic Probability Experiment – any process that generates one or more observable outcomes (Drawing a card) Event – Any outcome or set of outcomes in a sample (Drawing a king or queen) Probability Distribution – the probabilities of an event occurring in the sample space In your groups… Given a bag of 100 marbles contains 50 red, 30 blue, 10 yellow, and 10 green find the sample space, the probability distribution, and the probability a blue or green marble is drawn. What in this problem is the event and what is the experiment? Mutually Exclusive vs. Independent Events Mutually Exclusive Events- two events or experiments which have no outcomes in common P(A or B) = P(A) + P(B) Independent Events- occurrence or non occurrence of one event has no effect on probability of the other event (flipping a coin) P(A and B) = P(A) x P(B) Complement- set of all outcomes not contained in the event 1 - P A pair of dice is rolled. The events of rolling a 6 and of rolling a double have the outcome (3,3) in common. These two events are NOT mutually exclusive. A pair of dice is rolled. The events of rolling a 9 and of rolling a double have NO outcomes in common. These two events ARE mutually exclusive. Some other examples of independent events are: Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die. Choosing a marble from a jar AND landing on heads after tossing a coin. Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card. Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die. In your groups… Years of Service Number of Employees The chart on the right shows the data of a company with employees who have been with them for various periods of time. 0-4 157 5-9 89 If an employee is chosen at random, what is the probability that the employee has 4 or fewer years of service and 5 – 9 years of service? 10-14 74 15-19 63 Is this a mutually exclusive event or independent? 20-24 42 25-29 38 30-34 37 35-39 21 40-44 8 In your groups… The probability of winning a game is .1 or 10% suppose the game is played on two different occasions. What is the probability of winning both games? Losing both games? And winning once and losing once? Random Variables Random Variable- a function that assigns a number to each outcome in the sample space The expected value is the sum of all the numerical values times their respective probabilities. Example: Outcome 15 16 17 18 19 20 Probability 0.1 0.3 0.2 0.2 0.1 0.1 Expected Value = 15(.1) + 16(.3) + 17(.2) + 18(.2) + 19(.1) + 20(.1) = 17.2 In your groups… Calculate the probabilities of the sum of rolling two dice? Example: Rolling snake eyes has a probability of 1/36. What is the expected value of the experiment? The sample space for rolling two dice is as follows: Probability Density Functions Probability 0.6 0.4 Probability 0.2 0 Red Blue Yellow Green Probability Density Function- any graph which corresponds to a probability distribution For example the above graph is of our data from the marble problem. The area of each bar represents the probability of drawing that color marble. The sum of all the areas equals 1. Exit Problem… Given a $1 lottery ticket with distribution of winning : $0 = .882746 $3 = .06 $5 = .04 $20 = .005 $40 = .002 $100 =.0002 $400 = .00005 $2500 = .000004 What is the expected value of the lottery ticket and how much do you lose per play? HINT:: Loss per play would equal the price of the ticket minus the expected value. Homework Pgs 872-873 #9-10 (Independent Probability) #18 and # 20 (Expected Value) #23-26 (Probability Distribution, Expected Value, Density Function)