P (A) = 0.60, P (B) = 0.55, P (A B c ) = 0.30

Transcription

1 Probability Math 42 Test October, 20 Dr. Pendergrass On my honor, I have neither given nor received any aid on this work, nor am I aware of any breach of the Honor Code that I shall not immediately report. Pledged: Print Name: Do these problems in your homework notebook, according to the same guidelines used for homework. You may use your class notes, and the lecture slides and notes posted on our class web-page. No other resources are allowed. This test is due at the beginning of class on Monday, October 4, 20.. Events A and B in a probability space have the following probabilities: a Find P A B. b Find P B A. c Find P A c B c. d Find P A c B c. e Are A and B independent? a P A B 0.8 b P B A 0.0 c P A c B c 0.0 d P A c B c 0. e No. P A 0.60, P B 0., P A B c An all-seeing, all-knowing genie informs you that the Washington Nationals baseball team will go -48 at home next season i.e. wins, 48 losses for games played at their home stadium. If you go to 9 home games next year chosen at random, what is the probability that the Nationals lose each one? A seven-card hand is dealt from a well-shuffled deck of 2 ordinary playing cards. Find the probabilities of the following events: a P four of a kind, and three of another kind. b P three of a kind, and two pairs. c P at least one ace. d P straight. Note: a straight is cards in sequence. An ace can be either high or low in a straight. a Page of Question continues on next page...

2 Continuing question... Test Dr. Pendergrass b c d , or if you exclude the straight flushes, In medical testing, a false positive occurs when the test indicates that the disease is present, when in fact no disease is present. A false negative occurs when the test indicates that no disease is present, when in fact disease is present. Let D be the event that disease is present, + be the event that the test is positive, and be the event that the test is negative. Suppose that a certain test has false positive rate P + D c 0.00, and false negative rate P D Suppose also that the prevalence of the disease is relatively rare: P D 0 6. a If the test comes back positive, what is the probability that the patient has the disease? b If the test comes back negative, what is the probability that the patient does not have the disease? Note that P + D P D 0.999, and P D c P + D c a By Bayes Theorem b Again by Bayes Theorem P D + P D c P D P + D P D P + D + P D c P + D c P D c P D c P D P D + P D c P D c A bag of candy contains 6 Reese s Cups, and 9 Snickers bars. Roll a fair six-sided die, and if the outcome is k, draw k pieces of candy out of the bag. a Find the probability that you get at least one Reese s Cup. b Let Y be the number of Reese s Cups you get. Find the probability mass function for Y. Page 2 of Question continues on next page...

3 Continuing question... Test Dr. Pendergrass a Let X be the result of the die toss, and condition on the value of X: P Y 0 6 P X j P Y 0 X j j j 6 j j Hence, P Y > 0 2/ b For k {, 2,, 4,, 6} we have P Y k 6 6 P X j P Y k X j jk k j k The mass function for Y is tabulated below: k jk j P Y k approx Urn I contains a one-dollar bill, Urn II contains two one-dollar bills and a ten-dollar bill, and Urn III contains three one-dollar bills, two ten-dollar bills, and a hundred-dollar bill. But the urns aren t labeled, so you don t know which is which. Choose an urn at random, and then pick a bill at random from that urn. Let X be the value of the bill you get. a Find the probability mass function for X. b Suppose that X. What is the probability that the bill came from Urn I? a Let W be the number of the selected urn, and condition on W : P X x P W w P X x W w w P X x W w w if x 4 8 if x 0 8 if x 00 if x if x 0 if x 00 Page of Question 6 continues on next page...

4 Continuing question 6... Test Dr. Pendergrass b P W, X P W X P X P W P X W P X 8 6. Compute the probability that a -card poker hand is void in at least one suit. Consider the complementary event that a -card hand has at least one card from each suit. Then it must have exactly two cards from one suit, and exactly one card from each of the remaining suits. The probability of this is So the probability of being void in at least on suit is Alternate Number the suites, for example as follows: Hearts, 2 Clubs, Spades, 4 Diamonds. Let E i be the event that the hand is void in suite i, for i, 2,, and 4. We need to find P 4 i E i. We will use inclusion/exclusion. Note that Hence, by inclusion/exclusion 9 P E i 2, i 4 26 P E i E j 2, i < j 4 P E i E j E k 2, i < j < k 4 P E E 2 E E 4 0 P ie i If there are 2 strangers in a room, what is the probability that no two of them celebrate their birthday in the same month? Clearly state the basic assumptions you are making. Assume that each person is equally likely to be born in any of the twelve months. Then the probability that none of 2 individuals share the same birth-month is 2!/ Page 4 of

5 Math 42 Test Dr. Pendergrass 9. Reorder the digits through 9 randomly. Call a reordering deranged if digit d does not appear in place d, for all d, 2,, 9. So for instance, is not deranged, since the digit 2 appears in place 2; but the reordering is deranged. Find the probability that a reordering is deranged. This is the Matching Problem with N 9. A deranged reordering corresponds to the situation in the Matching Problem in which no man gets his own hat. The probability of this is 9 n n! n0 0. A man has n keys, one of which will open his door. a If he tries the keys at random, discarding those that do not work, what is the probability that he will open the door on his k th try? b What is the probability if he does not discard previously tried keys? a Assume that all the n! orderings of the keys are equally likely. The door is opened on the k th try if and only if the correct key is in position k in the reordering. The probability of this is n! n!, k, 2,,, n n b If previous keys are not discarded, then on any try, the correct key is chosen with probability /n, and an incorrect key is chosen with probability /n. It follows that the probability that the door opens on the k th try is k,, k, 2,, n n Page of