(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "X Window System Mathematica Notebook Front End Version 2.2"; X11StandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, fullFontName, -9, "-*-times-bold-r-normal-*-*-240-75-75-*-*-*-*", 24, fontName, "times"; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, fullFontName, -9, "-*-times-bold-r-normal-*-*-180-75-75-*-*-*-*", 18, fontName, "times"; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, fullFontName, -9, "-*-times-medium-i-normal-*-*-140-75-75-*-*-*-*", 14, fontName, "times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, fullFontName, -9, "-*-times-bold-r-normal-*-*-180-75-75-*-*-*-*", 18, fontName, "times"; 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fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, fullFontName, -9, "-*-times-medium-i-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = leftheader, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, fullFontName, -9, "-*-times-medium-i-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = leftfooter, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "courier"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times";paletteColors = 128; automaticGrouping; currentKernel; ] :[font = subtitle; inactive; preserveAspect; fontName = "Times"; startGroup] Problem Set 1 :[font = subsection; inactive; preserveAspect; fontName = "Times"; startGroup] Problem 4 :[font = text; inactive; preserveAspect; fontName = "Times"] By choosing twenty numbers, the casino has essentially turned those balls "red" and left the other ones "white". The problem then is the same as choosing n red balls (without replacement) from a group of 20 red balls and 60 white balls. The probability of you choosing your first number from among the casinos group is 20/80; the second is 19/79, etc. Therefore the probability of choosing 6 is: :[font = input; preserveAspect; startGroup] prob6 = 20/80*19/79*18/78*17/77*16/76*15/75 :[font = output; output; inactive; preserveAspect; endGroup] 51/395395 ;[o] 51 ------ 395395 :[font = text; inactive; preserveAspect; fontName = "Times"] The probability of choosing exactly 5 of the casino's numbers is given by the probability of choosing the first 5 from among the casino's picked numbers and the last on from the casino's unpicked numbers times the number of combinations you can get of the picked and unpicked numbers: :[font = input; preserveAspect; startGroup] prob5 = 20/80*19/79*18/78*17/77*16/76*60/75 * 6!/5!/1! :[font = output; output; inactive; preserveAspect; endGroup] 1224/395395 ;[o] 1224 ------ 395395 :[font = text; inactive; preserveAspect; fontName = "Times"] Likewise for 4 and 3: :[font = input; preserveAspect; startGroup] prob4 = 20/80*19/79*18/78*17/77*60/76*59/75 * 6!/4!/2! :[font = output; output; inactive; preserveAspect; endGroup] 9027/316316 ;[o] 9027 ------ 316316 :[font = input; preserveAspect; startGroup] prob3 = 20/80*19/79*18/78*60/77*59/76*58/75 * 6!/3!/3! :[font = output; output; inactive; preserveAspect; endGroup] 10266/79079 ;[o] 10266 ----- 79079 :[font = text; inactive; preserveAspect; fontName = "Times"] The expected return is the payoff for each combination times the probability for that combination: :[font = input; preserveAspect; startGroup] expectedreturn=prob6*1500.+prob5*50.+prob4*6.+prob3*2. :[font = output; output; inactive; preserveAspect; endGroup] 0.7791259373537854 ;[o] 0.779126 :[font = text; inactive; preserveAspect; fontName = "Times"] You could have equally well asked how often the casino will chose from among your choices. This should give you the same results. For the probability that the casino chooses all 6 of your numbers: :[font = input; preserveAspect; startGroup] prob6c = 6/80 * 5/79 * 4/78 * 3/77 * 2/76 * 1/75 * 74/74 * 73/73 * 72/72 * 71/71 * 70/70 * 69/69 * 68/68 * 67/67 * 66/66 * 65/65 * 64/64 * 63/63 * 62/62 * 61/61 * 20!/6!/14! :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "prob6c" is similar to existing symbol "prob6". :[font = output; output; inactive; preserveAspect; endGroup] 51/395395 ;[o] 51 ------ 395395 :[font = text; inactive; preserveAspect; fontName = "Times"; endGroup; endGroup] Which is the same result as given above. The other probabilities will be the same also and thus the expected return will be the same. ^*)