5) An experiment consists of rolling a red die and a green die and noting the result of each roll. Find the events:

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1 1) A telephone sales representative makes successive calls to potential customers, and the result of each call is recorded as a sale (S) or no sale (N). Calls are continued until either 2 successive sales are made, 2 successive calls result in no sale, or a total of 3 calls is made. Find the sample space for the situation. SS (stop 2 successive sales) SNS (stop a total of 3 calls made) SNN (stop two successive calls result in no sale) NSS (stop 2 successive sales) NSN (stop a total of 3 calls made) NN (stop two successive calls result in no sale) Answer: {SS, SNS, SNN, NSS, NSN, NN} 3) An experiment with outcomes O 1, O 2, O 3, O 4, O 5 has an assignment of probabilities w 1, w 2, w 3, w 4, w 5. Suppose 0 2 is three times as likely as 0 4, w 1 =.40, and w 3 = w 4 = w 5. Find w 2. Let x = w 4 w 1 + w 2 + w 3 + w 4 + w 5 = x + x + x + x = x = 1 6x =.60 x =.10 w 2 = 3x =.30 5) An experiment consists of rolling a red die and a green die and noting the result of each roll. Find the events: E = {exactly one die has a result of 3} F = {at least one die has a result of 3} G = {the sum of the results is an odd number} H = {the sum of the results is at most 7}

2 E = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3)} F = (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3), (3, 3) G = (1, 2), (1, 4), (1, 6) (2, 1), (2, 3), (2, 5) (3, 2), (3, 4),(3, 6) (4, 1), (4, 3), (4, 5) (5, 2), (5, 4),(5, 6) (6, 1), (6, 3), (6, 5) H = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4) (4, 1), (4, 2), (4,3) (5, 1), (5, 2) (6, 1) 7) A purse contains 3 nickels, 3 dimes, and 2 quarters. Coins are withdrawn, one at a time, without replacement, until either one nickel has been selected or the total value of the coins selected is at least 35 cents. Find the sample space of the experiment. N (Stop one nickel selected) DN (Stop one nickel selected) DDN (Stop one nickel selected) DDDN (Stop one nickel selected) DDDQ (Stop at least 35 cents) DDQ (Stop at least 35 cents) DQ (Stop 35 cents) QN (Stop one nickel selected) QD (Stop 35 cents)

3 QQ (Stop at least 35 cents) Answer: S = {N, DN, DDN, DDDN, DDDQ, DDQ, DQ, QN, QD, QQ} 9) Suppose an experiment has 4 outcomes and the frequencies of the outcomes are 3 n, 11 n, 5 n, and 7. What does n have to be, for those frequencies to reflect n probabilities? Probabilities have to add to 1. The bottoms have to be the same as the tops added together = 26 Answer: n = 26 11) An experiment has outcomes O 1, O 2, and O 3 with Pr[{0 1, 0 2 }] =.9 Pr[{ 0 1, 0 3 }] =.85. What is Pr[O 2 ]? w 1 + w 2 + w 3 = 1 Pr[{ 0 1, 0 3 }] + w 2 =.85 w 2 =.15 13) An experiment with five outcomes has w 1 =.2, w 2 = w 3, w 4 =.35, and w 5 = w 1 + w 2. Find w 2. Let w 2 = x w 1 + w 2 + w 3 + w 4 + w 5 = x + x (.2 + x) = x + x x = 1 3x +.75 = 1 3x =.25 x =

4 15) A die is weighted so that outcomes 1 and 2 are equally likely; outcome 1 is three times as likely as each of outcomes 3, 4, 5, and 6; and these outcomes are equally likely. What probabilities (weights) should be assigned to the outcomes to reflect this information? Since outcome 1 is three times as likely as outcome 3 Let w 3 = x Then w 1 = 3x Since outcomes 1 and 2 are equally likely w 2 = 3x Since outcomes 3, 4, 5, and 6 are equally likely, w 4 = x w 5 = x w 6 = x 3x + 3x + x + x + x + x = 1 10x = 1 x = 1 10 w1 = w2 = 3 1 ; w3 = w4 = w5 = w6 = ) Suppose an experiment has three possible outcomes 0 1, 0 2, and 0 3. Also suppose that 0 1 occurs with frequency 1 a, outcome O 2 with frequency 2 a, and 0 3 with frequency 3. What probabilities should be assigned to the outcomes to a reflect this? That s another way of saying that O 2 is twice as likely to occur as O 1 and O 3 is three times as likely to occur as O 1 Let w 1 = x then w 3 = 3x and w 2 = 2x x + 2x + 3x = 1 6x = 1

5 x = 1 That means that the a in the problem is ) An unfair die has the property that when rolled each of the odd numbers is equally likely to land uppermost, each of the even numbers is equally likely to land uppermost, and each odd number is twice as likely to land up as an even number. The die is rolled and the result is noted. Find the probability that the results is in the event {1, 3, 6}. Odd number twice as likely to land uppermost means w 1 = 2w 2 Let all the even probabilities = x and all the odd probabilities = 2x 2x + x + 2x + x + 2x + x = 1 9x = 1 x = 1 9 {1, 3, 6} = w1 + w3 + w6 = ) Suppose an experiment has sample space S = {O 1, O 2, O 3, O 4, O 5 }. Also, let outcome O i be assigned probability w i where w 1 = w 2 = 1/5 w 3 = 2/5 w 4 = w 5 = 1/10. Find Pr[E] and Pr[F ], where E = {O 2, O 4 } and F = { O 1, O 4 }. Pr[E] = Pr[{O 2, O 4 }] = w 2 + w =

6 Pr[F ] = Pr[{O 2, O 3, O 5 }] = w 2 + w 3 = You could also do this by finding Pr[F] and then taking that away from 1. 23) In a finite mathematics class with 300 students, 10% withdraw, 15% receive a grade of A, 20% receive B, 35% get C, 15% get D, and 5% receive F. Assign probabilities to the events pass the course and withdraw or fail the course. Pass = A + B + C + D = = 85% = 0.85 Withdrawal or Fail = = 15% = ) A lottery is designed so that on average it pays $10 on 1 play out of 100, $1000 on 1 play out of 5000, and $10,000 on 1 play out of 100,000. All other plays have no return. Assign probabilities to the four possible outcomes of buying a lottery ticket. Pr[win 10] = 0.01 Pr[win 1000] = Pr[win 10000] = Add those and get Subtract from 1 to get not win = ) A product code consists of one number selected from {1, 2, 3, 5} followed by two letters, not necessarily distinct, selected from the set {Q, T, Z}. For example, 3QT is such a code. An experiment consists of selecting a code at random. How many codes are possible? What probability should be assigned to each code? Find the probability of the event that the code contains the number 3. 3 slots: 4 choices for first slot, 3 choices for second slot, 3 choices for third slot 4 * 3 * 3 = 36

7 Probability for each code should be 1 36 If the code contains the number 3 then the # of codes would b 1 * 3 * 3 = 9 The probability would be ) There are balls numbered 1 through 5 in a box. Two balls are selected at random in succession without replacement, and the number on each ball is noted. How many outcomes does this is experiment have? What probability should be assigned to each? What probabilities should be assigned to the event that at least one ball has an odd number? 5 * 4 = 20 outcomes What probability should be assigned to each? 1 20 What probabilities should be assigned to the event that at least one ball has an odd number? There are 3 odd numbers {1, 3, 5) and 2 even numbers {2, 4} Note the phrase at least We could draw an odd and then a different odd: 3 * 2 = 6 We could draw and odd and then an even: 3 * 2 = 6 We could draw and even and then an odd: 2 * 3 = = 18 and ) A fair coin is flipped three times, and the result (heads or tails) is noted after each toss. How many outcomes does this experiment have? What probability should be assigned to each? Outcomes: 2 * 2 * 2 = 8 The probability that should be assigned to each = 1 8

8 33) In a large law school class, the Professor decides to call on students by using their initials. Thus, Dirty Harry is called DH and Mary Contrary is called MC. A colleague from the math faculty notes that at least two students will have the same initials. How many students are in the class? Total # of combinations: 26 * 26 = 676 For a guaranteed repeat we need 677 students 35) A box contains three balls: 1 red, 1 blue, and 1 yellow. The balls are drawn at random one after the other until all have been removed from the box. The color of each ball is noted as it is drawn. How many outcomes are there for this experiment? Outcomes: 3 * 2 * 1 = 6 37) There are balls numbered 1 through 5 in a box. One ball is selected at random, replaced, and then a second ball is drawn. The number on each ball is noted. How many outcomes does this is experiment have? What probability should be assigned to each? What probabilities should be assigned to the event that at least one ball has an odd number? Outcomes: 5 * 5 = 25 The probability that should be assigned to each = 1 25 What probabilities should be assigned to the event that at least one ball has an odd number? There are 3 odd numbers {1, 3, 5) and 2 even numbers {2, 4} Note the phrase at least We could draw an odd and then a different odd: 3 * 3 = 9 We could draw and odd and then an even: 3 * 2 = 6 We could draw and even and then an odd: 2 * 3 = = 21 Probability is ) A psychology experiment consists of presenting a stimulus to a subject and noting the response. At each presentation of the stimulus, the subject is evaluated as making a strong association (S), a weak association (W), or no

9 association (N). If the subject makes a strong association on the first presentation, the experiment ends. Otherwise it continues and a second stimulus is presented. The experiment ends after the second stimulus if the subject makes a strong association; otherwise it continues and a third stimulus is presented. The experiment ends after the response to the third stimulus (if one is presented) is recorded. An outcome is a sequence of responses to stimuli. a) Find the event that there is a strong association. b) Find the event that there is at least one weak association. c) Find the event that there is exactly one weak association. Outcomes: Strong (stop, first one strong) Weak, Strong (stop, second one strong) None, Strong (stop, second one strong) Weak, Weak, Strong Weak, Weak, Weak Weak, Weak, None Weak, None, Strong Weak, None, Weak Weak, None, None None, Weak, Strong None, Weak, Weak None, Weak, None None, None, Strong None, None, Weak None, None, None

10 The event that there is a strong association: {S, WS, NS, WWS, WNS, NWS, NNS} The event that there is at least one weak association: {WS, WWS, WWW, WWN, WNS, WNW, WNN, NWS, NWW, NWN, NNW} The event that there is exactly one weak association: {WS, WNS, WNN, NWS, NWN, NNW}