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Worksheet: Conditional Probability

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V63.0233, Theory of ProbabilityName:Worksheet for Sections 3.1–3.2 : Introduction to Conditional ProbabilityJuly 7, 20091.If A is the event that an astronaut is a member of the armed services, T is the event thathe or she was once a test pilot, and S is the event that he or she is a scientist, express each of thefollowing probabilities symbolically:(i) the probability that an astronaut who was once a test pilot is a member of the armed services(ii) the probability that an astronaut who is a member of the armed services is a scientist butwas never a test pilot(iii) the probability that an astronaut who is not a scientist was once a test pilot(iv) the probability that an astronaut who is a member of the armed services but was never apilot is a scientist2.There are 60 qualiﬁed applicants for teaching positions in a high school, of which some havehad at least ﬁve years’ teaching experience and some have not, some are married and some aresingle, with the exact breakdown beingMarriedSingleAt least ﬁve years teach-126ing experienceLessthanﬁveyears2418teaching experienceIf the order in which the applicants are interviewed by the principal is random, M is the event thatthe ﬁrst applicant interviewed is married, and F is the event that the ﬁrst applicant interviewedwill have had at least ﬁve years’ teaching experience, determine the following probabilities from thetable:(i) P (M )(iv) P (F )(viii) P (M | F )(v) P (M ∩ F )(ii) P (M )(ix) P (F | M )(vi) P (M | F )(iii) P (F )(vii) P (F | M )(x) P (M ∩ F )1Appendix CLife TableNumber of survivors at single years of Age, out of 100,000 Born Alive, by Race and Sex: United States, 1990.All races All races Age Both sexes Male Female Age Both sexes Male Female 0 100000 100000 100000 43 94707 92840 96626 1 99073 98969 99183 44 94453 92505 96455 2 99008 98894 99128 45 94179 92147 96266 3 98959 98840 99085 46 93882 91764 96057 4 98921 98799 99051 47 93560 91352 95827 5 98890 98765 99023 48 93211 90908 95573 6 98863 98735 99000 49 92832 90429 95294 7 98839 98707 98980 50 92420 89912 94987 8 98817 98680 98962 51 91971 89352 94650 9 98797 98657 98946 52 91483 88745 94281 10 98780 98638 98931 53 90950 88084 93877 11 98765 98623 98917 54 90369 87363 93436 12 98750 98608 98902 55 89735 86576 92955 13 98730 98586 98884 56 89045 85719 92432 14 98699 98547 98862 57 88296 84788 91864 15 98653 98485 98833 58 87482 83777 91246 16 98590 98397 98797 59 86596 82678 90571 17 98512 98285 98753 60 85634 81485 89835 18 98421 98154 98704 61 84590 80194 89033 19 98323 98011 98654 62 83462 78803 88162 20 98223 97863 98604 63 82252 77314 87223 21 98120 97710 98555 64 80961 75729 86216 22 98015 97551 98506 65 79590 74051 85141 23 97907 97388 98456 66 78139 72280 83995 24 97797 97221 98405 67 76603 70414 82772 25 97684 97052 98351 68 74975 68445 81465 26 97569 96881 98294 69 73244 66364 80064 27 97452 96707 98235 70 71404 64164 78562 28 97332 96530 98173 71 69453 61847 76953 29 97207 96348 98107 72 67392 59419 75234 30 97077 96159 98038 73 65221 56885 73400 31 96941 95962 97965 74 62942 54249 71499 32 96800 95785 97887 75 60557 51519 69376 33 96652 95545 97804 76 58069 48704 67178 34 96497 95322 97717 77 55482 45816 64851 35 96334 95089 97624 78 52799 42867 62391 36 96161 94843 97525 79 50026 39872 59796 37 95978 94585 97419 80 47168 36848 57062 38 95787 94316 97306 81 44232 33811 54186 39 95588 94038 97187 82 41227 30782 51167 40 95382 93753 97061 83 38161 27782 48002 41 95168 93460 96926 84 35046 24834 44690 42 94944 93157 96782 85 31892 21962 41230 Figure 1: Life table of Americans in 1990. Taken from Grinstead and Snell, Introduction to Proba-bility Theory3.In the Life Table, one ﬁnds that in a population of 100,000 females, 89.835% can expect tolive to age 60, while 57.062% can expect to live to age 80. Given that a woman is 60, what is theprobability that she lives to age 80? Repeat the problem for men.2

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