Math& 107, Summer 2016 Instructor: Marina McLeod Probability and Counting Make sure to show your work for each problem and give all probabilities as fractions or decimals rounded to 3 digits. Unless otherwise specified, \"die\" or \"dice\" refer to standard, 6-sided dice. (1) For each procedure described below: (i) Give/describe a natural sample space. (ii) Give an example of an event with a probability that is neither 0 nor 1 that is not in the sample space (not a simple event). Try to give the event in a \"natural\" way, rather than just listing different simple events. (iii) Is every event in your sample space equally likely? (a) (b) (c) (d) Flip one penny and one nickel. Roll one red and one green die. Record both rolls. Pick a person at random and record her or his height in inches. Pick a person at random and record whether he or she is currently employed part-time, full-time, or not at all. (e) From a bag containing 2 apples, 4 oranges, and 1 banana, pick one piece of fruit. Answer each of the following. Make sure to specify when you use the counting strategies we have learned: multiplication rule, factorials, permutations, and/or combinations. (2) How many hands of 7 cards can be made from a deck of 52 cards? (3) If there are 10 people on a committee, how many different ways could they appoint a Chair, Secretary, and Treasurer from their committee? (4) There are 81 senators in the National Congress of Brazil: 3 from each of the 26 states, and 3 from one federal district. (a) How many subcommittees of 27 people can be selected from these senators? (b) How many subcommittees of 27 people can be selected if there must be one senator from each state and federal district? Calculate the probabiliy of each event below. (5) Procedure: Draw a card from a standard, shuffled deck (52 cards total). Event 1: 8 Event 3: 5 or spade Event 2: 8 or King Event 4: a red card or an even number (6) Procedure: A bag contains: 2 red balls, 5 green balls, and 8 blue balls. Draw 3 balls, with replacement. Event 1: green, red, blue Event 3: all 3 same color Event 2: 3 greens Event 4: all different colors (7) Same as number (6), but without replacement. (8) A rental car company has the shown number of vehicles available in each category. Procedure: The agent at the counter selects a random vehicle for you to rent. 2 5 7 Passenger Passenger Passenger Compact 4 1 0 Economy 0 9 0 SUV 0 5 3 Minivan 0 0 2 Use the notation for conditional probability when applicable. Event 1: SUV Event 2: 7 Passenger Event 3: SUV, if you specify that you want a 5 passenger vehicle Event 4: 7 Passenger, if you already know it's a minivan Event 5: 5 Passenger or SUV Event 6: 2 Passenger and Economy Event 7: not a 7 passenger SUV (9) Consider the following game: You pay $2 to play. Then you roll a die and flip a coin. You win money according to the given chart (you get the highest amount that your result matches). Find the expected value for this game. Be careful not to count any events twice. Result Payout 6 and H $15 even roll $3 T $1 anything else $0 (10) There are 100 Starburst candies in a bag: 25 each of orange, red, pink, and yellow. (a) How many ways can 20 people pick a single Starburst from the bag? (We can distinguish between the 4 colors, but not between individual Starbursts of one color.) (b) How many ways can those 20 people pick Starbursts so that the first 5 people all get red? (c) Consider P (the first 5 people get red). (i) Calculate this probability using the multiplication rule for probability (look at what you did for number (7).) (ii) Now calculate this probability using your results from parts (a) and (b). (iii) Why are the two different, and which one is correct