document.write( "Question 1195827: probability of rain on day one is 50% and if it rain on day 1 probability of rain on day 2 is 20% and if it do not rain on day 1 probability or rain on day 2 is also 20% find the probability of it rained on first day given it rain on second day
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Algebra.Com's Answer #828394 by ikleyn(52781)\"\" \"About 
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\n" ); document.write( "(1) probability of rain on day one is 50%, and
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document.write( "Let A be the event \"there is a rain on day 1\".\r\n" );
document.write( "Let B be the event \"there is a rain on day 2\".\r\n" );
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document.write( "We are given \r\n" );
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document.write( "    (1)  P(A) = 0.5;\r\n" );
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document.write( "    (2)  P(B|A) = 0.2;\r\n" );
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document.write( "    (3)  P(B|notA) = 0.2.\r\n" );
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document.write( "    +---------------------------------+\r\n" );
document.write( "    |    They want you find P(A|B).   |\r\n" );
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document.write( "From (1) and (2), we have  P(A ∩ B) = P(B|A)*P(A) = 0.2*0.5 = 0.1.\r\n" );
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document.write( "From (1) and (3), we have P(notA ∩ B) = P(B|notA)*P(notA) = 0.2*0.5 = 0.1.\r\n" );
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document.write( "Hence,  P(B) = P(A ∩ B) + P(notA ∩ B) = 0.1 + 0.1 = 0.2.\r\n" );
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document.write( "It implies  P(A|B) = P(A ∩ B)/P(B) = 0.1/0.2 = 1/2 = 0.5 = 50%.\r\n" );
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document.write( "So, the solution is formally complete and the answer is \r\n" );
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document.write( "    |   \"the probability of it rained on first day  |\r\n" );
document.write( "    |      given it rains on second day is 0.5\".    |\r\n" );
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\n" ); document.write( "\n" ); document.write( "        It is a formal solution, but the informal solution is  OBVIOUS:\r
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document.write( "Indeed, from the given part, the probability that it rains the second day \r\n" );
document.write( "is 20% independently on what was happened on the 1st day.\r\n" );
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document.write( "So, the fact that \"it is given 20% probability of the rain in the 2nd day\" does not add any new information.\r\n" );
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document.write( "Hence, the only condition (1) works to predict the weather at the 1st day,\r\n" );
document.write( "and it says that it is 50% probability of rain, giving the answer to the problem's question.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved and explained - - - both formally and informally.\r
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