Chapter 3. Probability

84 Chapter 3: ProbabilityCombination of three things, taken from 10 things10! 10 ∗9∗8∗ 7! 10 ∗ 9 ∗87203C 10 = == = = 120(10 − 3)!3! 7!3! 3∗2 ∗16b. Probability that all three selected are defectiveP(all three are defective)= P(first defective) ∗ P(second defective) ∗ P(third defective)=0.3 ∗ 0.3 ∗ 0.3= 0.02713. Age Discrimination32! 32 ∗31∗30∗ 29 ∗ 28! 863,040Number of possible selections= 32 C 4 = == = 35, 960(32 − 4)! 4! 28! 4!24Only one way four oldest can be selected135,960P(four oldest were selected out of 32)= = 0. 0000278This probability is certainly low enough to suspect that the hospital may have discriminated against olderphysicians since by chance, this outcome of selecting the four oldest physicians would have happened about 3times in 100,000 times.14. Computer DesignByte= 8 bits, each as 0 or 1, Find number of possible bytesNumber of sequences of eight bits, each with one of two possible outcomes= 2 8 = 256Are there enough different bytes to represent the characters we use? It would certainly seem to be. On thetypical keyboard, there would be 26 alphabetical characters, 10 numbers, and 16 other characters for a total of52 characters. This would leave many bytes left to represent many other characters not listed above.15. Selection of Treatment GroupsSix subjects selected from a pool of 15 volunteersFirst find number of possible selections in the form of combinations of 6 things selected from 15 things, so wefind 15 C 615C615! 15∗14∗13∗12∗11∗10∗ 9! 15 ∗14∗13∗14∗13∗12∗11∗10= ===(15 − 6)!6!9!6!6 ∗ 5∗4 ∗3∗2 ∗13,603,600= 5,005720There is only one way out of 5,005 samples that the six youngest can be selected15005P(six youngest are selected)= = 0. 000200Should this happen, it would appear to be highly likely the sample was selected based on age rather than atrandom since this would have happened about twice out of 10,000 samples.16. Air RoutesNumber of routes of three cities= 3!= 3 ∗ 2 ∗ 1= 6 different routesThese would be: ABC, ACB, BAC, BCA, CAB, and CBANumber of routes of eight cities= 8!= 40,320We won’t take the time or space to list all 40,320 of these.17. Social Security Numbers Nine digits in SS number, each can be 0-1011011,000,000,000P(of a given SS number)= =0. 0000000019=