Section 7.1 Experiments, Sample Spaces, and Events Experiments An experiment is an activity with observable results. 1. Which of the follow are experiments? (a) Going into a room and turning on a light. (b) Rolling a die and observing the number that appears on the uppermost face. (c) Flipping a coin and observing wether the coin lands on heads or tails. (d) Driving a car up and down the street. Sample Spaces and Events A sample space, S, isthesetconsistingofallpossibleoutcomesinanexperiment.thisis like the universal set from Section 6.1. An event, E, isasubsetofthesamplespace. Note: Every set operation discussed in Section 6.1 stillappliesinthiscurrentsection,i.e. the universal set U is replaced by the sample space S, andtheterm subset isreplacedwiththe term event. 2. Let S = {d, g, k, q, v, y} be a sample space of an experiment and let E = {d, g}, F = {d, q, y}, and G = {g, k, v} be events of this experiment. Find the events E c and F c \ G. 3. Consider the sample space S = {s, t, n}. How many total events are there for this sample space?
4. Let S = {5, 9, 12} be a sample space associated with an experiment. (a) List all events of this experiment. (b) How many subsets of S contain the number 12? # of subsets = 4 (c) How many subsets of S contain the number 9 or the number 12? # of Subsets = 6 5. An experiment consists of spinning the hand of the numbered disc shown in the following figure and then observing the region in which the pointer stops. (If the needle stops on a line, the result is discounted and the needle is spun again. (a) What is the appropriate sample space S for this experiment? (b) Describe the event E the spinner points to the number 2. (c) Describe the event F the spinner points to an odd number. 2 Fall 2017, Maya Johnson
6. An experiment consists of tossing a coin and observing the side that lands up and then rolling a fair 4-sided die and observing the number rolled. Let H and T represent heads and tails respectively. (a) Describe the sample space S corresponding to this experiment. (b) What is the event E 1 that an even number is rolled? (c) What is the event E 2 that a head is tossed or a 3 is rolled? (d) What is the event E 3 that a tail is tossed and an odd number is rolled? 7. The numbers 3, 4, 5, and 7 are written on separate pieces of paper and put into a hat. Two pieces of paper are drawn at the same time and the product of the numbers is recorded. Find the sample space. 3 Fall 2017, Maya Johnson
8. An opinion poll is conducted among a state s electorate to determine the relationship between their income levels and their stands on a proposition aimed at reducing state income taxes. Voters are classified as belonging to either the low-, middle-, or upper-income group. They are asked whether they favor, oppose, or are undecided about the proposition. Let the letters L, M, andu represent the low-, middle-, and upper-income groups, respectively, and let the letters f, o, and u represent their responsesfavor, oppose, and undecided, respectively. (a) Describe a sample space S corresponding to this poll. ( M,f ), ( Mio), ( Mu), ( V,f ), ( U, ),( U,u ) } (b) Describe the event E 1 that a respondent favors the proposition. - E, = { ( Lift, ( Nf ), ( UA ) } (c) Describe the event E 2 that a respondent opposes the proposition and does not belong to the low-income group. af ( Md,( 40 ) } 4 Fall 2017, Maya Johnson
9. Two fair 6-sided dice are rolled and the numbers shown uppermost are observed. Find the number of outcomes in the following events. (a) The sum of the numbers is 7. (b) A5isrolled. (c) A2isrolledorthesumofthediceisnomorethan5. 5 Fall 2017, Maya Johnson
10. A jar contains 8 marbles numbered 1 through 8. An experiment consists of randomly selecting a marble from the jar, observing the number drawn, and then randomly selecting a card from a standard deck and observing the suit of the card (hearts, diamonds, clubs, or spades). (a) How many outcomes are in the sample space for this experiment? (b) How many outcomes are in the event an even number is drawn? (c) How many outcomes are in the event a number more than 1 is drawn and a red card is drawn? (d) How many outcomes are in the event a number less than 2 is drawn or a club is not drawn? 6 Fall 2017, Maya Johnson