document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207265>Question 1207265</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An urn contains 7 red marbles, 6 white marbles, and 8 blue marbles marbles. A child randomly selects three (without replacement) from the urn. Round to four decimal places.\r<BR>\n" );
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document.write( "Find the probability all three marbles are the same color. <BR>\n" );
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document.write( "Find the probability that none of the three marbles are white. </SPAN>\n" );
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document.write( "Part 1\r<BR>\n" );
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document.write( "Let's find the probability of getting 3 red marbles in a row.<BR>\n" );
document.write( "7 red out of 7+6+8 = 21 total<BR>\n" );
document.write( "7/21 = probability of getting a red marble<BR>\n" );
document.write( "6/20 = probability of a second red marble (no replacement)<BR>\n" );
document.write( "5/19 = probability of a third red marble (no replacement)<BR>\n" );
document.write( "Note the countdown of numerator and denominator. These countdowns are because of the no replacement. <BR>\n" );
document.write( "(7/21)*(6/20)*(5/19) = 1/38 is the probability of selecting 3 red marbles in a row (no replacement)<BR>\n" );
document.write( "Another approach is to use the nCr combination formula to find there are 7C3 = 35 ways to pick the 3 red marbles out of 21C3 = 1330 ways to select 3 marbles. Then 35/1330 = 1/38\r<BR>\n" );
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document.write( "Now calculate the probability of getting 3 white marbles in a row.<BR>\n" );
document.write( "6 white out of 21 total<BR>\n" );
document.write( "6/21, 5/20, 4/19 are the probabilities of 1st,2nd,3rd marbles as white if there's no replacement.<BR>\n" );
document.write( "(6/21)*(5/20)*(4/19) = 2/133 is the probability of selecting 3 white marbles in a row with no replacement.\r<BR>\n" );
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document.write( "Lastly we'll find the probability of getting 3 blue marbles in a row.<BR>\n" );
document.write( "8 blue out of 21 total<BR>\n" );
document.write( "8/21, 7/20, 6/19 are the probabilities of 1st,2nd,3rd marbles as blue where there's no replacement.<BR>\n" );
document.write( "(8/21)*(7/20)*(6/19) = 4/95 is the probability of selecting 3 blue marbles in a row with no replacement.\r<BR>\n" );
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document.write( "To summarize so far<BR>\n" );
document.write( "P(3 red) = 1/38<BR>\n" );
document.write( "P(3 white) = 2/133<BR>\n" );
document.write( "P(3 blue) = 4/95<BR>\n" );
document.write( "Add up those fractions to get 111/1330 = <font color=red>0.0835</font> as the approximate probability of getting 3 of the same color in a row.<BR>\n" );
document.write( "Note that those 3 probability values we added represent mutually exclusive events. \r<BR>\n" );
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document.write( "Part 2\r<BR>\n" );
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document.write( "7 red + 8 blue = 15 red or blue out of 7+6+8 = 21 marbles total<BR>\n" );
document.write( "15/21, 14/20, 13/19 are the probability values for 1st,2nd,3rd marbles as non-white if no replacements are made.\r<BR>\n" );
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document.write( "Multiply out those fractions to get 13/38 = <font color=red>0.3421</font> approximately.\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "Probability all three marbles are the same color.<BR>\n" );
document.write( "<font color=red>0.0835</font>\r<BR>\n" );
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document.write( "Probability that none of the three marbles are white.<BR>\n" );
document.write( "<font color=red>0.3421</font>\r<BR>\n" );
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document.write( "Both results are approximate.<BR>\n" );
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