document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=476720>Question 476720</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a group of 50 men, 16 are bald, 20  have beards, and 4 are bald and have beards. A man is selected at random. (a) find the probability he is bald given he has a beard. (b) find the probability he has a beard given he is bald.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #326883 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20060)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=326883\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In a group of 50 men, 16 are bald, 20  have beards, and 4 are bald and have beards. A man is selected at random. (a) find the probability he is bald given he has a beard. (b) find the probability he has a beard given he is bald.<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
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document.write( "The other tutor got the correct answers in decimal form, \r\n" );
document.write( "but didn't give the slightest clue as to how he got them.\r\n" );
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document.write( "Make this chart, or an equivalent Venn diagram:\r\n" );
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document.write( "         |   Bald   |    Non-Bald   |   Totals\r\n" );
document.write( "Beard    |          |               |     \r\n" );
document.write( "----------------------------------------------\r\n" );
document.write( "Beardless|          |               |\r\n" );
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document.write( "Totals   |          |               |     \r\n" );
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document.write( "We can fill in the 50 for the total men in the bottom \r\n" );
document.write( "right, 16 for the total bald, 20 for total with\r\n" );
document.write( "beards and 4 bald and bearded, straight from the\r\n" );
document.write( "given numbers, like this:\r\n" );
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document.write( "         |   Bald   |    Non-Bald   |   Totals\r\n" );
document.write( "Bearded  |     4    |               |     20\r\n" );
document.write( "----------------------------------------------\r\n" );
document.write( "Beardless|          |               |\r\n" );
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document.write( "Totals   |    16    |               |     50\r\n" );
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document.write( "Next by subtraction we get that \r\n" );
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document.write( "1. Left column: there are 16-4 or 12 beardless bald men.\r\n" );
document.write( "2. Top row: there are 20-4 = 16 bearded non-bald.\r\n" );
document.write( "3. Bottom row: there are 50-16 = 34 total non-bald men.\r\n" );
document.write( "4. Right column: there are 50-20 = 30 total beardless men \r\n" );
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document.write( "         |   Bald   |    Non-Bald   |   Totals\r\n" );
document.write( "Bearded  |     4    |       16      |     20\r\n" );
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document.write( "Beardless|    12    |               |     30\r\n" );
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document.write( "Totals   |    16    |       34      |     50\r\n" );
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document.write( "Finally we can fill in the number of beardless \r\n" );
document.write( "non-bald men either of two ways either by \r\n" );
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document.write( "1. Middle column: 34-16 = 18 \r\n" );
document.write( "or\r\n" );
document.write( "2. Middle row: 30-12 = 18.\r\n" );
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document.write( "         |   Bald   |    Non-Bald   |   Totals\r\n" );
document.write( "Bearded  |     4    |       16      |     20\r\n" );
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document.write( "Beardless|    12    |       18      |     30\r\n" );
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document.write( "Totals   |    16    |       34      |     50\r\n" );
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document.write( "Now we can choose either to do the conditional \r\n" );
document.write( "probability problems by:\r\n" );
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document.write( "1.  Reduced sample space mthod\r\n" );
document.write( "2.  By the formula method\r\n" );
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document.write( "Let's do it first by the reduced sample space method.\r\n" );
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document.write( "(a)  find the probability he is bald given he has a beard. \r\n" );
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document.write( "Since we are given that he has a beard. we can eliminate\r\n" );
document.write( "everything except the top row, so we just look only at \r\n" );
document.write( "this part of the above chart:\r\n" );
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document.write( "         |   Bald   |    Non-Bald   |   Totals\r\n" );
document.write( "Bearded  |     4    |       16      |     20\r\n" );
document.write( "----------------------------------------------\r\n" );
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document.write( "Using this reduced sample space we see that the\r\n" );
document.write( "desired conditional probability \r\n" );
document.write( "P(bald|bearded) = 4/20 = 1/5\r\n" );
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document.write( "(b) find the probability he has a beard given he is bald.\r\n" );
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document.write( "Since we are given that he is bald, we can eliminate\r\n" );
document.write( "everything except the left column, so we just look at \r\n" );
document.write( "this part of the chart above:\r\n" );
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document.write( "         |   Bald   |   \r\n" );
document.write( "Bearded  |     4    |     \r\n" );
document.write( "---------------------\r\n" );
document.write( "Beardless|    12    |   \r\n" );
document.write( "---------------------\r\n" );
document.write( "Totals   |    16    |   \r\n" );
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document.write( "Using this reduced sample space we see that the\r\n" );
document.write( "desired conditional probability \r\n" );
document.write( "P(bearded|bald) = 4/16 = 1/4\r\n" );
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document.write( "                                                 P(X and Y)\r\n" );
document.write( "We can also do these by this formula:  P(X|Y) = ————————————\r\n" );
document.write( "                                                    P(Y)\r\n" );
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document.write( "From the entire chart:\r\n" );
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document.write( "         |   Bald   |    Non-Bald   |   Totals\r\n" );
document.write( "Bearded  |     4    |       16      |     20\r\n" );
document.write( "----------------------------------------------\r\n" );
document.write( "Beardless|    12    |       18      |     30\r\n" );
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document.write( "Totals   |    16    |       34      |     50\r\n" );
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document.write( "(a)   \r\n" );
document.write( "                    P(bald and bearded)    4/50      4     1\r\n" );
document.write( "  P(bald|bearded) = ——————————————————— = —————— = ———— = ———\r\n" );
document.write( "                         P(bearded)        20/50    20     5\r\n" );
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document.write( "(b)   \r\n" );
document.write( "                    P(bearded and bald)    4/50      4     1\r\n" );
document.write( "  P(bearded|bald) = ——————————————————— = —————— = ———— = ———\r\n" );
document.write( "                           P(bald)         16/50    16     4\r\n" );
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document.write( "Either method for conditional probability is correct.\r\n" );
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document.write( "Edwin</PRE> </SPAN>\n" );
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