document.write( "Question 548416: Given that P(A) = 0.5, P (B) =0.3, and P(A and B) = 0.10, determine P(AlB) \n" ); document.write( "
Algebra.Com's Answer #356987 by Edwin McCravy(20055) $\"\"$ $\"About$
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document.write( "Two ways to do it: (1) By formula and  (2) By chart:  \r\n" );
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document.write( "(1) By formula.\r\n" );
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document.write( "This is the easy way when you don't have complement event A' or \r\n" );
document.write( "the complement event  B'\r\n" );
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document.write( "P(A|B) =  =  = \r\n" );
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document.write( "However, you may have other problems with A' or with B'.  If so the\r\n" );
document.write( "chart method is better:\r\n" );
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document.write( "(2) By chart\r\n" );
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document.write( "P(A) = 0.5, P (B) =0.3, and P(A and B) = 0.10, determine P(AlB)\r\n" );
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document.write( "Make this probability chart:\r\n" );
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document.write( "      |   B  |  B' | Totals \r\n" );
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document.write( "   A  |      |     |   \r\n" );
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document.write( "   A' |      |     |\r\n" );
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document.write( "Totals|      |     |   \r\n" );
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document.write( "Fill in the lower right cell total with probability 1.0, (for certainty).\r\n" );
document.write( "Fill in the upper left cell with 0.1 because that cell is to contain\r\n" );
document.write( "the probability of (A and B)\r\n" );
document.write( "Fill in the total probability of A as 0.5\r\n" );
document.write( "Fill in the total probability of B as 0.3 \r\n" );
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document.write( "      |   B  |  B' | Totals\r\n" );
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document.write( "   A  |  0.1 |     |   0.5\r\n" );
document.write( "   A' |      |     |\r\n" );
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document.write( "Totals|  0.3 |     |   1.0\r\n" );
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document.write( "Fill in the probability for (A and B') by subtracting 0.1 from 0.5, getting 0.4\r\n" );
document.write( "Fill in the probability for (A' and B) by subtracting 0.1 from 0.3, getting 0.2\r\n" );
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document.write( "      |   B  |  B' | Totals\r\n" );
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document.write( "   A  |  0.1 | 0.4 |   0.5\r\n" );
document.write( "   A' |  0.2 |     |\r\n" );
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document.write( "Totals|  0.3 |     |   1.0\r\n" );
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document.write( "Fill in the total probability for A' by subtracting 0.5 from 1.0, getting 0.5\r\n" );
document.write( "Fill in the total probability for B' by subtracting 0.3 from 1.0, getting 0.7\r\n" );
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document.write( "      |   B  |  B' | Totals\r\n" );
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document.write( "   A  |  0.1 | 0.4 |   0.5\r\n" );
document.write( "   A' |  0.2 |     |   0.5\r\n" );
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document.write( "Totals|  0.3 | 0.7 |   1.0\r\n" );
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document.write( "Finally, fill in the probability for A' and B' by subtracting 0.4 from 0.7, getting 0.3.  Notice that it automatically checks because the total of B'\r\n" );
document.write( "is correctly 0.7:\r\n" );
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document.write( "      |   B  |  B' | Totals\r\n" );
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document.write( "   A  |  0.1 | 0.4 |   0.5\r\n" );
document.write( "   A' |  0.2 | 0.3 |   0.5\r\n" );
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document.write( "Totals|  0.3 | 0.7 |   1.0\r\n" );
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document.write( "Now we can find any of the 8 possible conditional probabilities:\r\n" );
document.write( "In particular, we want to find P(A given B), since B is given, we ignore everything except the B column, the numbers in red below:\r\n" );
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document.write( "      |   B  |  B' | Totals\r\n" );
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document.write( "   A  |  0.1 | 0.4 |   0.5\r\n" );
document.write( "   A' |  0.2 | 0.3 |   0.5\r\n" );
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document.write( "Totals|  0.3 | 0.7 |   1.0\r\n" );
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document.write( "The desired conditional probability P(A|B) =  = \r\n" );
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document.write( "The advantage of the table method is that we can easily find all 8 possible\r\n" );
document.write( "conditional probabilities, by ignoring everything except what is given, and\r\n" );
document.write( "dividing the probability by the total in that column or row.  For instance,\r\n" );
document.write( "suppose we wanted to find\r\n" );
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document.write( "P(B|A).  \r\n" );
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document.write( "We would ignore everything but the (given) A row \r\n" );
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document.write( "      |   B  |  B' | Totals\r\n" );
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document.write( "   A  |  0.1 | 0.4 |   0.5\r\n" );
document.write( "   A' |  0.2 | 0.3 |   0.5\r\n" );
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document.write( "Totals|  0.3 | 0.7 |   1.0\r\n" );
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document.write( "and divide the probability \r\n" );
document.write( "for B in that row by the total probability of A, getting  = .\r\n" );
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document.write( "All 8 of the conditional probabilities can be gotten from the table:\r\n" );
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document.write( "      |   B  |  B' | Totals\r\n" );
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document.write( "   A  |  0.1 | 0.4 |   0.5\r\n" );
document.write( "   A' |  0.2 | 0.3 |   0.5\r\n" );
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document.write( "Totals|  0.3 | 0.7 |   1.0\r\n" );
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document.write( "P(A|B) =  = \r\n" );
document.write( "P(A|B') =  = \r\n" );
document.write( "P(A'|B) =  = \r\n" );
document.write( "P(A'|B') =  = \r\n" );
document.write( "P(B|A) =  = \r\n" );
document.write( "P(B|A') =  = \r\n" );
document.write( "P(B'|A) =  = \r\n" );
document.write( "P(B'|A') =  =   \r\n" );
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document.write( "---------------------------\r\n" );
document.write( "Edwin
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