document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194145>Question 1194145</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a 6/35 lottery game, a player pays $1 and selects six numbers from 1 to 35. Any player who has chosen the six winning numbers wins $1,000,000. Assuming that this is the only way to win, what is the expected value of this game? (round to the nearest cent) </SPAN>\n" );
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document.write( "We have n = 35 numbers to pick from and r = 6 slots to fill.<BR>\n" );
document.write( "The order doesn't matter. \r<BR>\n" );
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document.write( "Use the nCr combination formula.<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "35 C 6 = (35!)/(6!*(35-6)!)<BR>\n" );
document.write( "35 C 6 = (35!)/(6!*29!)<BR>\n" );
document.write( "35 C 6 = (35*34*33*32*31*30*29!)/(6!*29!)<BR>\n" );
document.write( "35 C 6 = (35*34*33*32*31*30)/(6!)<BR>\n" );
document.write( "35 C 6 = (35*34*33*32*31*30)/(6*5*4*3*2*1)<BR>\n" );
document.write( "35 C 6 = (1,168,675,200)/(720)<BR>\n" );
document.write( "35 C 6 = 1,623,160\r<BR>\n" );
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document.write( "There are 1,623,160 different combinations possible. <BR>\n" );
document.write( "This is the number of possible ways to play the lotto game.\r<BR>\n" );
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document.write( "Of this, there's only one way to win the grand prize since you must get all 6 matching numbers. Again the order doesn't matter. <BR>\n" );
document.write( "The odds of winning are 1,623,160 to 1 or about 1.6 million to 1. \r<BR>\n" );
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document.write( "We write that probability as the fraction 1/(1,623,160) or 1/1623160<BR>\n" );
document.write( "Type that fraction into your calculator to get the approximate decimal of 0.00000061608221 which is very small. There are six \"0\"s between the decimal point and the digit \"6\".\r<BR>\n" );
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document.write( "The probability of winning is roughly 0.00000061608221\r<BR>\n" );
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document.write( "The probability of losing the lotto game is roughly <BR>\n" );
document.write( "1-0.00000061608221 = 0.9999993839178\r<BR>\n" );
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document.write( "Make a table showing the two outcomes and their probabilities<BR>\n" );
document.write( "X = net pay out = amount the person walks away with<BR>\n" );
document.write( "X is either -1 or 999,999<BR>\n" );
document.write( "The person either loses $1, or wins $1 million to walk away with 1,000,000-1 = $999,999<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr><tr><TD >-1</TD><TD >0.9999993839178</TD></tr><tr><TD >999,999</TD><TD >0.00000061608221</TD></tr>\n" );
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document.write( "We'll form a new column called X*P(X)<BR>\n" );
document.write( "This is where we multiply each X and P(X) across the rows<BR>\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr><tr><TD >-1</TD><TD >0.9999993839178</TD><TD >-0.9999993839178</TD></tr><tr><TD >999,999</TD><TD >0.00000061608221</TD><TD >0.6160815939178</TD></tr>\n" );
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document.write( "Now add the values in that third column to get the expected value<BR>\n" );
document.write( "-0.9999993839178 + 0.6160815939178 = -0.38391779<BR>\n" );
document.write( "that rounds to -0.38\r<BR>\n" );
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document.write( "The player is expected to lose on average 38 cents per game. <BR>\n" );
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