document.write( "Question 1161740: On the leeward side of the island of Oahu, in a small village, about 72% of the residents are of Hawaiian ancestry. Let n = 1, 2, 3, … represent the number of people you must meet until you encounter the first person of Hawaiian ancestry in the village.\r
\n" ); document.write( "\n" ); document.write( "(a) Write out a formula for the probability distribution of the random variable n. (Enter a mathematical expression.)\r
\n" ); document.write( "\n" ); document.write( "P(n) =\r
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\n" ); document.write( "\n" ); document.write( "(b) Compute the probabilities that n = 1, n = 2, and n = 3. (For each answer, enter a number. Round your answers to three decimal places.)
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\n" ); document.write( "\n" ); document.write( "(c) Compute the probability that n ≥ 4. Hint: P(n ≥ 4) = 1 − P(n = 1) − P(n = 2) − P(n = 3). (Enter a number. Round your answer to three decimal places.)\r
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\n" ); document.write( "\n" ); document.write( "(d) What is the expected number of residents in the village you must meet before you encounter the first person of Hawaiian ancestry? Hint: Use μ for the geometric distribution and round. (Enter a number. Round your answer to the nearest whole number.) \n" ); document.write( "

Algebra.Com's Answer #785354 by ikleyn(52908) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " I re-ordered your questions in a way which seems to be more logical to me . . . \r
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\n" ); document.write( "\n" ); document.write( "On the leeward side of the island of Oahu, in a small village, about 72% of the residents are of Hawaiian ancestry.
\n" ); document.write( "Let n = 0, 1, 2, 3, … represent the number of people you must meet until you encounter the first person of Hawaiian ancestry in the village.\r
\n" ); document.write( "\n" ); document.write( "(a) Compute the probabilities that n = 1, n = 2, and n = 3. (For each answer, enter a number. Round your answers to three decimal places.)
\n" ); document.write( "P(0) =
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\n" ); document.write( "P(3) =\r
\n" ); document.write( "\n" ); document.write( "(b) Write out a formula for the probability distribution of the random variable n. (Enter a mathematical expression.)
\n" ); document.write( "P(n) =\r
\n" ); document.write( "\n" ); document.write( "(c) Compute the probability that n ≥ 4. Hint: P(n ≥ 4) = 1 − P(n=1) − P(n=2) − P(n=3). (Enter a number. Round your answer to three decimal places.)\r
\n" ); document.write( "\n" ); document.write( "(d) What is the expected number of residents in the village you must meet before you encounter the first person of Hawaiian ancestry?
\n" ); document.write( "Hint: Use μ for the geometric distribution and round. (Enter a number. Round your answer to the nearest whole number.)\r
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\n" ); document.write( "\n" ); document.write( "Solution\r
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document.write( "(a)  P(0) = is the probability that the FIRST person you encounter is of Hawaiian ancestry = 0.72.\r\n" );
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document.write( "     P(1) = is the probability that the first person you encounter is not of Hawaiian ancestry, \r\n" );
document.write( "                but the SECOND person you encounter is of Hawaiian ancestry = (1-0.72)*0.72 = 0.2016.\r\n" );
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document.write( "     P(2) = is the probability that the first TWO persons you encounter are not of Hawaiian ancestry,\r\n" );
document.write( "               but the THIRD person you encounter is of Hawaiian ancestry = (1-0.72)*(1-0.72)*0.72 = 0.056448.\r\n" );
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document.write( "     P(3) = is the probability that the first THREE persons you encounter are not of Hawaiian ancestry,\r\n" );
document.write( "               but the FOURTH person you encounter is of Hawaiian ancestry = (1-0.72)*(1-0.72)*(1-0.72)*0.72 = 0.015805.\r\n" );
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document.write( "(b)  P(n) = is the probability that the first \"n\" persons you encounter are not of Hawaiian ancestry,\r\n" );
document.write( "               but the (n+1)-th person you encounter is of Hawaiian ancestry = .\r\n" );
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document.write( "            Obviously, the values P(0), P(1), P(2), P(3) form a geometric progression starting from 0.72 with the common ratio of (1-0.72) = 0.28.\r\n" );
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document.write( "(c)  P(n>=4) = 1 - P(1) - P(2) - P(3) = 1 - 0.72 - 0.2016 - 0.056448 - 0.015805 = 0.006147.\r\n" );
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document.write( "(d)  This Math expectation is\r\n" );
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document.write( "     M = 0*P(0) + 1*P(1) + 2*P(2) + 3*P(3) + . . . =  +  +  + . . . =\r\n" );
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document.write( "       =  = \r\n" );
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document.write( "             (see the formula at https://math.stackexchange.com/questions/1269008/formula-for-r2r23r3-nrn)\r\n" );
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document.write( "       =  =  = 0.388888...\r\n" );
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\n" ); document.write( "\n" ); document.write( "All questions are answered.\r
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