document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203260>Question 1203260</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Alonzo, Bob, and Casper work bussing tables at a restaurant. Alonzo has a 60% chance, Bob has a 10% chance, and Casper has a 30% chance of bussing tables in the middle area of the restaurant. If Alonzo is bussing tables, he has a 6% chance of breaking a dish. If Bob is bussing tables, he has a 2% chance of breaking a dish. Finally, if Casper is bussing tables, he has a 4% chance of breaking a dish. If there is a broken dish in the middle of the restaurant, what is the probability it was broken by Casper?\r<BR>\n" );
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document.write( "Multiple Choice<BR>\n" );
document.write( "0.012<BR>\n" );
document.write( "0.040<BR>\n" );
document.write( "0.240<BR>\n" );
document.write( "0.720 </SPAN>\n" );
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document.write( "Answer: <font color=red size=4>Choice C) 0.240</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "There are at least two methods we can use. \r<BR>\n" );
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document.write( "Method 1\r<BR>\n" );
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document.write( "A = Alonzo buses tables in the middle area<BR>\n" );
document.write( "B = Bob buses tables in the middle area<BR>\n" );
document.write( "C = Casper buses tables in the middle area<BR>\n" );
document.write( "D = Dish is broken in the middle area<BR>\n" );
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document.write( "From the second sentence of the instructions, we are given these 3 pieces of info:<BR>\n" );
document.write( "P(A) = 0.60<BR>\n" );
document.write( "P(B) = 0.10<BR>\n" );
document.write( "P(C) = 0.30\r<BR>\n" );
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document.write( "The next 3 sentences offer these additional 3 pieces of info:<BR>\n" );
document.write( "P(D given A) = 0.06<BR>\n" );
document.write( "P(D given B) = 0.02<BR>\n" );
document.write( "P(D given C) = 0.04\r<BR>\n" );
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document.write( "<font color=red>Goal: Compute P(C given D)</font><BR>\n" );
document.write( "In other words, we want to find P(C) when we know event D has occurred. \r<BR>\n" );
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document.write( "Use <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Bayes%27_theorem\">Bayes' Theorem</a><BR>\n" );
document.write( "P(C given D) = P(D given C)*P(C)/P(D)\r<BR>\n" );
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document.write( "We already have the values of P(D given C) and P(C)\r<BR>\n" );
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document.write( "We'll need P(D)\r<BR>\n" );
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document.write( "Use the <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Law_of_total_probability\">Law of Total Probability</a><BR>\n" );
document.write( "P(D) = P(D and A) + P(D and B) + P(D and C)<BR>\n" );
document.write( "P(D) = P(D given A)*P(A) + P(D given B)*P(B) + P(D given C)*P(C)<BR>\n" );
document.write( "P(D) = 0.06*0.60 + 0.02*0.10 + 0.04*0.30<BR>\n" );
document.write( "P(D) = 0.05\r<BR>\n" );
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document.write( "We have enough info to compute the following<BR>\n" );
document.write( "P(C given D) = P(D given C)*P(C)/P(D)<BR>\n" );
document.write( "P(C given D) = 0.04*0.30/0.05<BR>\n" );
document.write( "P(C given D) = 0.012/0.05<BR>\n" );
document.write( "P(C given D) = 12/50<BR>\n" );
document.write( "P(C given D) = 6/25<BR>\n" );
document.write( "P(C given D) = <font color=red size=4>0.240</font>\r<BR>\n" );
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document.write( "This is why the final answer is <font color=red size=4>choice C</font>\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "Let's say that there are 1000 instances of a single dish being bused.\r<BR>\n" );
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document.write( "Alonzo handles 60% of them, Bob does 10%, Casper does the remaining 30%<BR>\n" );
document.write( "60% of 1000 = 0.60*1000 = 600<BR>\n" );
document.write( "10% of 1000 = 0.10*1000 = 100<BR>\n" );
document.write( "30% of 1000 = 0.30*1000 = 300\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "Alonzo buses 600 times<BR>\n" );
document.write( "Bob buses 100 times<BR>\n" );
document.write( "Casper buses 300 times\r<BR>\n" );
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document.write( "Alonzo breaks a dish 6% of the time<BR>\n" );
document.write( "6% of 600 = 0.06*600 = 36<BR>\n" );
document.write( "If Alonzo buses 600 times then we expect him to break around 36 dishes.\r<BR>\n" );
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document.write( "Bob breaks a dish 2% of the time<BR>\n" );
document.write( "2% of 100 = 0.02*100 = 2<BR>\n" );
document.write( "If Bob buses 100 times then we expect him to break around 2 dishes.\r<BR>\n" );
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document.write( "Casper breaks a dish 4% of the time<BR>\n" );
document.write( "4% of 300 = 0.04*300 = 12<BR>\n" );
document.write( "If Casper buses 300 times then we expect him to break around 12 dishes.\r<BR>\n" );
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document.write( "An total estimate of 36+2+12 = 50 dishes have been broken.<BR>\n" );
document.write( "P(D) = probability of a broken dish<BR>\n" );
document.write( "P(D) = (50 dishes broken)/(1000 dishes total)<BR>\n" );
document.write( "P(D) = 50/1000<BR>\n" );
document.write( "P(D) = 0.05<BR>\n" );
document.write( "This was calculated earlier in the previous section.\r<BR>\n" );
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document.write( "Of those 50 broken dishes, Casper is estimated to break 12 of them.<BR>\n" );
document.write( "12/50 = <font color=red>0.240</font><BR>\n" );
document.write( "If we encounter a broken dish, then there's a 24% chance that Casper did it.<BR>\n" );
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