Chapter 13

Examples.1. Roll a die. Your chance of getting either a “1” or a “6” is_________2. A box contains 2 red marbles and 3 yellow ones. Youreach in and choose a marble at random (withoutlooking).red red yellow yellow yellowa) The chance the marble is red is _______b) The chance the marble is yellow is __________

4. I have a box containing 1 red die, 2 green dice and awhite die. I reach in and choose a die at random, and Iroll it.a) What is the chance I get a green die?b) What is the chance I don’t get the green die?c) What is the chance I get the number “6”?d) What is the chance I don’t get the number “6”?e) What is the chance I get a green die and the number“6”?f) What is the chance I get a green die or the number “6”(or both)?

5. I toss a coin 2 times.a) What is the chance it lands “H” both times? Write this asHH.b) What is the chance it lands HT?c) What is the chance it lands TH?d) What is the chance it lands TT?e) What is the chance we get at least one H?

Drawing from a box: think of a box of tickets that areidentical except that they have a number written onthem or they are different colors.The box is shaken so the tickets are thoroughly mixed. Wereach in and choose a ticket “at random”. We mightreach in again and choose a second ticket. There aretwo ways we can do this:• with replacement (we look at the first ticket and thenwe put it back in the box and mix up the tickets beforechoosing the second ticket)• without replacement (we do not put the first ticketback)

The Multiplication Rule

7. A ticket will be drawn at random from Box A or Box B.Box A:Box B:You win $1 if the ticket is black. Which box is better foryou?

8. Two tickets will be drawn at random without replacementfrom Box A or Box B.Box A:Box B:You win $1 if both tickets are red. Which box is better foryou?

9. A girl reaches in to the following box with her left handand draws out a marble at random, then reaches inwith her right hand and draws out a second marble atrandom (without replacing the first one).red blue greena) What is the chance she gets the red marble in her lefthand?b) What is the chance she gets the green marble in herright hand?c) What’s the chance she gets the red in her left handand the green in her right hand?

10. An elementary school class has 12 boys and 13 girls.Two children are chosen at random to go to lunch with theteacher.a) What is the chance that both of the children are girls?b) What is the chance that both of the children are boys?c) What is the chance that the first is a girl and thesecond is a boy?d) What is the chance that the first is a boy and thesecond is a girl?

10. (continued)e) What is the chance that one of the children is a girl andthe other is a boy?f) What is the chance the children have the same gender?g) What is the chance that at least one of the children is agirl?

11. A box contains 2 red marbles and 3 yellow ones.red red yellow yellow yellowCase A: You reach in and choose two marbles withoutreplacement.a) The chance the first marble is red is _______b) The chance the second marble is red is _______c) The chance the first marble is red and the second marbleis yellow is:d) The chance the first marble is yellow and the secondmarble is red is:e) The chance one marble is yellow and the other marble isred is:

12. A box contains 2 red marbles and 3 yellow ones:red red yellow yellow yellowCase B: You reach in and choose two marbles withreplacement.a) The chance the first marble is red is _______b) The chance the second marble is red is _______c) The chance the first marble is red and the second marbleis yellow is:d) The chance the first marble is yellow and the secondmarble is red is:e) The chance one marble is yellow and the other marble isred is:

IndependenceTwo things are independent if the chances for the secondare the same, no matter how the first turns out.Examples:• Coin tosses are independent of each other• Rolls of a die are independent of each other• Draws from a box are independent if we are drawingwith replacement• Draws from a box are NOT independent if we aredrawing with replacement• Cards in a poker hand are not independente.g. Toss 2 coins. Are the following things independent?A: getting a H on the first tossB: getting two H’s

13. a) One draw will be made from the following box:1 1 2 1 1 2Are color and number independent?b) As above, but for the box:1 1 2 1 1 1 2 22c) As above, but for the box:1 1 2 1 1 1 1 22

The Multiplication Rule for IndependentEvents

14. A coin is tossed 5 times. What is the chance of gettingHTTHT? What about HHHHH?15. Roll a die and toss a coin. What is the chance that thedie lands 6 and the coin lands T?

16. A girl draws marbles with replacement from thefollowing box:red blue greena) What is the chance she gets the red marble first?b) What is the chance she gets the green marblesecond?c) What’s the chance she gets the red marble first andthe green marble second?d) If she draws 6 times, what is the chance all six drawsare red?e) If she draws 6 times, what is the chance that at leastone of the six draws is red?

The Collins CaseA white girl with blond hair in a ponytail was seen leavingthe scene of a crime in a yellow car with an AfricanAmerican man with a beard and mustache.• yellow car 1/10• man with mustache 1/4• girl with ponytail 1/10• girl with blond hair 1/3• African American man with beard 1/10• interracial couple in car 1/1000Multiply them all together and we get 1/12,000,000.A couple was identified with all these characteristics, andthe prosecution argued that since only 1 in 12 millioncouples could satisfy these conditions, the couple musthave been the criminals.WHAT’S WRONG WITH THIS REASONING?