document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178107>Question 1178107</A>: <I><SPAN STYLE=\"word-break: break-all;\"> During christmas mr queen mendoza collects coins for his nephew and her nieces . He collected 15 25 centavo coins , 30 pcs 1 peso coins , 20 pieces 5 pesos and 5 pieces 10 peso coin. He place all those coins in a bowl and let his or her nephews and nieces get coins as much as their one hand can hold .<BR>\n" );
document.write( " 1. Suppose her to use old ms juvy can hold 4 coins at a time, what is the expected number of 1 peso coin that her niece could get? <BR>\n" );
document.write( "2. Suppose her two years old niece gemma gets 3 different coins. What is the expected amount of the money that her niece could get?\r<BR>\n" );
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document.write( "Thank you for the answer with complete solution and explanation for I can better understand the way it is solved </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850358 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850358\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down these probability problems.\r<BR>\n" );
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document.write( "**1. Expected Number of 1 Peso Coins (4 Coins Total)**\r<BR>\n" );
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document.write( "* **Total Coins:** 15 (25 centavos) + 30 (1 peso) + 20 (5 pesos) + 5 (10 pesos) = 70 coins<BR>\n" );
document.write( "* **Number of 1 Peso Coins:** 30<BR>\n" );
document.write( "* **Number of Non-1 Peso Coins:** 70 - 30 = 40<BR>\n" );
document.write( "* **Niece Takes:** 4 coins\r<BR>\n" );
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document.write( "To calculate the expected number of 1 peso coins, we need to consider the probabilities of getting 0, 1, 2, 3, or 4 of them.\r<BR>\n" );
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document.write( "* **Probability of getting x 1 peso coins:**<BR>\n" );
document.write( "    * P(X = x) = [C(30, x) * C(40, 4 - x)] / C(70, 4)<BR>\n" );
document.write( "    * Where C(n, k) is the combination formula (n choose k).\r<BR>\n" );
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document.write( "Let's calculate:\r<BR>\n" );
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document.write( "* C(70, 4) = 70! / (4! * 66!) = 916895<BR>\n" );
document.write( "* C(30, 0) = 1; C(40, 4) = 91390<BR>\n" );
document.write( "* C(30, 1) = 30; C(40, 3) = 9880<BR>\n" );
document.write( "* C(30, 2) = 435; C(40, 2) = 780<BR>\n" );
document.write( "* C(30, 3) = 4060; C(40, 1) = 40<BR>\n" );
document.write( "* C(30, 4) = 27405; C(40, 0) = 1\r<BR>\n" );
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document.write( "Now calculate the probabilities:\r<BR>\n" );
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document.write( "* P(X = 0) = (1 * 91390) / 916895 ≈ 0.0997<BR>\n" );
document.write( "* P(X = 1) = (30 * 9880) / 916895 ≈ 0.3235<BR>\n" );
document.write( "* P(X = 2) = (435 * 780) / 916895 ≈ 0.3705<BR>\n" );
document.write( "* P(X = 3) = (4060 * 40) / 916895 ≈ 0.1772<BR>\n" );
document.write( "* P(X = 4) = (27405 * 1) / 916895 ≈ 0.0299\r<BR>\n" );
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document.write( "Expected value (E[X]):\r<BR>\n" );
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document.write( "* E[X] = (0 * 0.0997) + (1 * 0.3235) + (2 * 0.3705) + (3 * 0.1772) + (4 * 0.0299)<BR>\n" );
document.write( "* E[X] = 0 + 0.3235 + 0.741 + 0.5316 + 0.1196 = 1.7157\r<BR>\n" );
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document.write( "Alternatively, since she picks 4 coins out of 70, and 30 are 1 peso coins, then the expected number of 1 peso coins is:\r<BR>\n" );
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document.write( "* E[X] = (30/70) * 4 = 120/70 = 12/7 ≈ 1.714\r<BR>\n" );
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document.write( "So the expected number of 1 peso coins is approximately 1.71.\r<BR>\n" );
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document.write( "**2. Expected Amount of Money (3 Different Coins)**\r<BR>\n" );
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document.write( "* **Possible Combinations of 3 Different Coins:**<BR>\n" );
document.write( "    * 25 centavos, 1 peso, 5 pesos<BR>\n" );
document.write( "    * 25 centavos, 1 peso, 10 pesos<BR>\n" );
document.write( "    * 25 centavos, 5 pesos, 10 pesos<BR>\n" );
document.write( "    * 1 peso, 5 pesos, 10 pesos\r<BR>\n" );
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document.write( "* **Values of Combinations:**<BR>\n" );
document.write( "    * 0.25 + 1 + 5 = 6.25 pesos<BR>\n" );
document.write( "    * 0.25 + 1 + 10 = 11.25 pesos<BR>\n" );
document.write( "    * 0.25 + 5 + 10 = 15.25 pesos<BR>\n" );
document.write( "    * 1 + 5 + 10 = 16 pesos\r<BR>\n" );
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document.write( "* **Probabilities of Combinations:**<BR>\n" );
document.write( "    * P(25c, 1p, 5p) = (15/70) * (30/69) * (20/68) ≈ 0.0268<BR>\n" );
document.write( "    * P(25c, 1p, 10p) = (15/70) * (30/69) * (5/68) ≈ 0.0067<BR>\n" );
document.write( "    * P(25c, 5p, 10p) = (15/70) * (20/69) * (5/68) ≈ 0.0046<BR>\n" );
document.write( "    * P(1p, 5p, 10p) = (30/70) * (20/69) * (5/68) ≈ 0.0094\r<BR>\n" );
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document.write( "* **Expected Value:**<BR>\n" );
document.write( "    * E[Amount] = (6.25 * 0.0268) + (11.25 * 0.0067) + (15.25 * 0.0046) + (16 * 0.0094)<BR>\n" );
document.write( "    * E[Amount] ≈ 0.1675 + 0.0754 + 0.07015 + 0.1504 = 0.46345\r<BR>\n" );
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document.write( "Therefore the expected amount of money is approximately 0.46 pesos.<BR>\n" );
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