document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=287820>Question 287820</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 6 numbers are chosen at random out of 49 integers {1,2,3......,49} what is the probability that at least two of the integers are the same? Can you please provide a brief method as well. Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #208577 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=208577\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The probability that none of the numbers are the same is equal to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49%2F49+%2A+48%2F49+%2A+47%2F49+%2A+46%2F49+%2A+45%2F49+%2A+44%2F49 ALIGN=MIDDLE  ALT=\"49%2F49+%2A+48%2F49+%2A+47%2F49+%2A+46%2F49+%2A+45%2F49+%2A+44%2F49\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.727414103 ALIGN=MIDDLE  ALT=\".727414103\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The probability that at least two of them will be the same is equal to 1 minus the probability that none are the same.\r<BR>\n" );
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document.write( "This equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+.727414103 ALIGN=MIDDLE  ALT=\"1+-+.727414103\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.272585897 ALIGN=MIDDLE  ALT=\".272585897\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "To see how this works, try the same type of problem with smaller numbers.\r<BR>\n" );
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document.write( "Assume the numbers in the pot are [1,2,3,4].\r<BR>\n" );
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document.write( "Assume you want to draw 3 times.\r<BR>\n" );
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document.write( "What is the probability that none of the numbers are the same?\r<BR>\n" );
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document.write( "That probability would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F4+%2A+3%2F4+%2A+2%2F4 ALIGN=MIDDLE  ALT=\"4%2F4+%2A+3%2F4+%2A+2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2F64 ALIGN=MIDDLE  ALT=\"24%2F64\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F8 ALIGN=MIDDLE  ALT=\"3%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "On your first draw, you can get 4 out of 4 = 4/4.\r<BR>\n" );
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document.write( "On your second draw, you can get 3 out of 4 = 3/4.   That's because once you picked a number, you do not want to pick it again, so the remaining choices are 3 out of 4.\r<BR>\n" );
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document.write( "On your third draw, you can get 2 out of 4 = 2/4.   That's because once you picked 2 numbers, you do not want to pick either of those again, so the remaining choices are 2 out of 4.\r<BR>\n" );
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document.write( "The probability that you will not get any number being the same as any other number is therefore:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F4+%2A+3%2F4+%2A+2%2F4 ALIGN=MIDDLE  ALT=\"4%2F4+%2A+3%2F4+%2A+2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2F64 ALIGN=MIDDLE  ALT=\"24%2F64\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F8 ALIGN=MIDDLE  ALT=\"3%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The probability that you will pick at least 2 numbers that are the same will be equal to 1 minus the probability of none being the same which will equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+3%2F8 ALIGN=MIDDLE  ALT=\"1+-+3%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F8 ALIGN=MIDDLE  ALT=\"5%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "To see how this works, we'll detail out the possibilities.\r<BR>\n" );
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document.write( "The numbers in the pot are [1,2,3,4].\r<BR>\n" );
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document.write( "you will draw 3 from the pot randomly.\r<BR>\n" );
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document.write( "The pot is always the same on each draw.\r<BR>\n" );
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document.write( "The possible sets of 3 you can get are:\r<BR>\n" );
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document.write( "111<BR>\n" );
document.write( "112<BR>\n" );
document.write( "113<BR>\n" );
document.write( "114<BR>\n" );
document.write( "121<BR>\n" );
document.write( "122<BR>\n" );
document.write( "123<BR>\n" );
document.write( "124<BR>\n" );
document.write( "131<BR>\n" );
document.write( "132<BR>\n" );
document.write( "133<BR>\n" );
document.write( "134<BR>\n" );
document.write( "141<BR>\n" );
document.write( "142<BR>\n" );
document.write( "143<BR>\n" );
document.write( "144<BR>\n" );
document.write( "211<BR>\n" );
document.write( "212<BR>\n" );
document.write( "213<BR>\n" );
document.write( "214<BR>\n" );
document.write( "221<BR>\n" );
document.write( "222<BR>\n" );
document.write( "223<BR>\n" );
document.write( "224<BR>\n" );
document.write( "231<BR>\n" );
document.write( "232<BR>\n" );
document.write( "233<BR>\n" );
document.write( "234<BR>\n" );
document.write( "241<BR>\n" );
document.write( "242<BR>\n" );
document.write( "243<BR>\n" );
document.write( "244<BR>\n" );
document.write( "311<BR>\n" );
document.write( "312<BR>\n" );
document.write( "313<BR>\n" );
document.write( "314<BR>\n" );
document.write( "321<BR>\n" );
document.write( "322<BR>\n" );
document.write( "323<BR>\n" );
document.write( "324<BR>\n" );
document.write( "331<BR>\n" );
document.write( "332<BR>\n" );
document.write( "333<BR>\n" );
document.write( "334<BR>\n" );
document.write( "341<BR>\n" );
document.write( "342<BR>\n" );
document.write( "343<BR>\n" );
document.write( "344<BR>\n" );
document.write( "411<BR>\n" );
document.write( "412<BR>\n" );
document.write( "413<BR>\n" );
document.write( "414<BR>\n" );
document.write( "421<BR>\n" );
document.write( "422<BR>\n" );
document.write( "423<BR>\n" );
document.write( "424<BR>\n" );
document.write( "431<BR>\n" );
document.write( "432<BR>\n" );
document.write( "433<BR>\n" );
document.write( "434<BR>\n" );
document.write( "441<BR>\n" );
document.write( "442<BR>\n" );
document.write( "443<BR>\n" );
document.write( "444\r<BR>\n" );
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document.write( "That's a total of 64 possible sets of 3 you can draw from a pot of 4 consecutive numbers.\r<BR>\n" );
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document.write( "Out of that 64, the sets where all the numbers are different are:\r<BR>\n" );
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document.write( "123<BR>\n" );
document.write( "124<BR>\n" );
document.write( "132<BR>\n" );
document.write( "134<BR>\n" );
document.write( "142<BR>\n" );
document.write( "143<BR>\n" );
document.write( "213<BR>\n" );
document.write( "214<BR>\n" );
document.write( "231<BR>\n" );
document.write( "234<BR>\n" );
document.write( "241<BR>\n" );
document.write( "243<BR>\n" );
document.write( "312<BR>\n" );
document.write( "314<BR>\n" );
document.write( "321<BR>\n" );
document.write( "324<BR>\n" );
document.write( "341<BR>\n" );
document.write( "342<BR>\n" );
document.write( "412<BR>\n" );
document.write( "413<BR>\n" );
document.write( "421<BR>\n" );
document.write( "423<BR>\n" );
document.write( "431<BR>\n" );
document.write( "432\r<BR>\n" );
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document.write( "That's a total of 24 out of 64.\r<BR>\n" );
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document.write( "Out of that 64, the sets where at least 2 of the numbers are the same are:\r<BR>\n" );
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document.write( "111<BR>\n" );
document.write( "112<BR>\n" );
document.write( "113<BR>\n" );
document.write( "114<BR>\n" );
document.write( "121<BR>\n" );
document.write( "122<BR>\n" );
document.write( "131<BR>\n" );
document.write( "133<BR>\n" );
document.write( "141<BR>\n" );
document.write( "144<BR>\n" );
document.write( "211<BR>\n" );
document.write( "212<BR>\n" );
document.write( "221<BR>\n" );
document.write( "222<BR>\n" );
document.write( "223<BR>\n" );
document.write( "224<BR>\n" );
document.write( "232<BR>\n" );
document.write( "233<BR>\n" );
document.write( "242<BR>\n" );
document.write( "244<BR>\n" );
document.write( "311<BR>\n" );
document.write( "313<BR>\n" );
document.write( "322<BR>\n" );
document.write( "323<BR>\n" );
document.write( "331<BR>\n" );
document.write( "332<BR>\n" );
document.write( "333<BR>\n" );
document.write( "334<BR>\n" );
document.write( "343<BR>\n" );
document.write( "344<BR>\n" );
document.write( "411<BR>\n" );
document.write( "414<BR>\n" );
document.write( "422<BR>\n" );
document.write( "424<BR>\n" );
document.write( "433<BR>\n" );
document.write( "434<BR>\n" );
document.write( "441<BR>\n" );
document.write( "442<BR>\n" );
document.write( "443<BR>\n" );
document.write( "444\r<BR>\n" );
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document.write( "That's a total of 40 out of 64\r<BR>\n" );
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document.write( "The method that is good for drawing a set of 3 out of a set of 4 will be the same as the method for drawing a set of 6 out of a set of 49.\r<BR>\n" );
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document.write( "The method is:\r<BR>\n" );
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document.write( "To find the probability that at least 2 of the numbers are the same, you find the probability that none of the numbers are the same and you subtract that from 1.\r<BR>\n" );
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document.write( "When you draw a set of 3 out of 4 possible numbers, the probability that at least 2 of the numbers are the same will be:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+%28%284%2A3%2A2%29%2F%284%2A4%2A4%29%29 ALIGN=MIDDLE  ALT=\"1+-+%28%284%2A3%2A2%29%2F%284%2A4%2A4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+%2824%2F64%29 ALIGN=MIDDLE  ALT=\"1+-+%2824%2F64%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+%283%2F8%29 ALIGN=MIDDLE  ALT=\"1+-+%283%2F8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F8 ALIGN=MIDDLE  ALT=\"5%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "When you draw a set of 6 out of 49 possible numbers, the probability that at least 2 of the number are the same will be:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+%28%2849%2A48%2A47%2A46%2A45%2A44%29%2F%2849%2A49%2A49%2A49%2A49%2A49%29%29 ALIGN=MIDDLE  ALT=\"1+-+%28%2849%2A48%2A47%2A46%2A45%2A44%29%2F%2849%2A49%2A49%2A49%2A49%2A49%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That becomes:\r<BR>\n" );
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