document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203673>Question 1203673</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A softball team has three pitchers, A, B, and C, with winning percentages of 0,4, 0.6, and 0.8 respectively. These pitchers pitch with frequency 2,3 and 5 out of every 10 games, respectively In other words for a randomly selected game P(A) = 0.2 P(B) = 0.3, and P(C) = 0.5:<BR>\n" );
document.write( "A) Find P(team wins game) = P(W) <BR>\n" );
document.write( "B) P(A pitched game|team won) = P(A|W) </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "We have this information provided<BR>\n" );
document.write( "P(A) = 0.2<BR>\n" );
document.write( "P(B) = 0.3<BR>\n" );
document.write( "P(C) = 0.5<BR>\n" );
document.write( "and<BR>\n" );
document.write( "P(W given A) = 0.4<BR>\n" );
document.write( "P(W given B) = 0.6<BR>\n" );
document.write( "P(W given C) = 0.8\r<BR>\n" );
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document.write( "Something like \"P(W given A)\" means \"the probability of a win when we know 100% that pitcher A played\".<BR>\n" );
document.write( "P(W given A) = 0.4 can be rewritten to \"P(W) = 0.4 when pitcher A plays\"\r<BR>\n" );
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document.write( "This conditional probability formula<BR>\n" );
document.write( "P(A given B) = P(A and B)/P(B)<BR>\n" );
document.write( "rearranges to<BR>\n" );
document.write( "P(A and B) = P(A given B)*P(B)\r<BR>\n" );
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document.write( "Then use the law of total probability<BR>\n" );
document.write( "P(W) = P(W and A) + P(W and B) + P(W and C) <BR>\n" );
document.write( "P(W) = P(W given A)*P(A) + P(W given B)*P(B) + P(W given C)*P(C)<BR>\n" );
document.write( "P(W) = 0.4*0.2 + 0.6*0.3 + 0.8*0.5<BR>\n" );
document.write( "P(W) = <font color=red size=4>0.66</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "I'll use Bayes Theorem<BR>\n" );
document.write( "P(A given B) = P(B given A)*P(A)/P(B)\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "P(A given W) = P(W given A)*P(A)/P(W)<BR>\n" );
document.write( "P(A given W) = 0.4*0.2/0.66<BR>\n" );
document.write( "P(A given W) = 0.08/0.66<BR>\n" );
document.write( "P(A given W) = 8/66<BR>\n" );
document.write( "P(A given W) = <font color=red>4/33</font><BR>\n" );
document.write( "P(A given W) = <font color=red size=4>0.121212...</font> where the \"12\" repeats forever<BR>\n" );
document.write( "Round this however needed. \r<BR>\n" );
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document.write( "Another approach for Part (a)\r<BR>\n" );
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document.write( "Let's say the team plays 100 games.\r<BR>\n" );
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document.write( "Pitcher A does 20% of those games, so they pitch for 0.2*100 = 20 games.<BR>\n" );
document.write( "Pitcher B then does 0.3*100 = 30 games<BR>\n" );
document.write( "Pitcher C then does 0.5*100 = 50 games\r<BR>\n" );
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document.write( "Here's the table so far<BR>\n" );
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document.write( "<tr><TD ></TD><TD >Win</TD><TD >Loss</TD><TD >Total</TD></tr><tr><TD >A pitches</TD><TD ></TD><TD ></TD><TD >20</TD></tr><tr><TD >B pitches</TD><TD ></TD><TD ></TD><TD >30</TD></tr><tr><TD >C pitches</TD><TD ></TD><TD ></TD><TD >50</TD></tr><tr><TD >Total</TD><TD ></TD><TD ></TD><TD >100</TD></tr>\n" );
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document.write( "For the \"A pitches\" row, we'll have 8 wins and 12 losses.<BR>\n" );
document.write( "This is because pitcher A has a 40% success rate, so they win 0.4*20 = 8 games and lose the other 20-8 = 12 games.<BR>\n" );
document.write( "The other rows are filled out in a similar fashion.\r<BR>\n" );
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document.write( "We'll get this completed table<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Win</TD><TD >Loss</TD><TD >Total</TD></tr><tr><TD >A pitches</TD><TD >8</TD><TD >12</TD><TD >20</TD></tr><tr><TD >B pitches</TD><TD >18</TD><TD >12</TD><TD >30</TD></tr><tr><TD >C pitches</TD><TD >40</TD><TD >10</TD><TD >50</TD></tr><tr><TD >Total</TD><TD >66</TD><TD >34</TD><TD >100</TD></tr>\n" );
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document.write( "There are 66 wins out of 100 games total. <BR>\n" );
document.write( "Therefore, P(W) = 66/100 = <font color=red>0.66</font>\r<BR>\n" );
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document.write( "Another approach for Part (b)\r<BR>\n" );
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document.write( "Refer to the table in the previous section.\r<BR>\n" );
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document.write( "There are 66 wins. Of those wins, 8 are when pitcher A was in the game.<BR>\n" );
document.write( "P(A given W) = P(A and W)/P(W)<BR>\n" );
document.write( "P(A given W) = 8/66<BR>\n" );
document.write( "P(A given W) = <font color=red>4/33</font><BR>\n" );
document.write( "P(A given W) = <font color=red>0.121212...</font><BR>\n" );
document.write( "We focus on the \"win\" column because we know 100% that a win occurred. We ignore the loss column.<BR>\n" );
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