document.write( "Question 1200991: A local club sells 100 tickets every week from which a winning ticket is randomly selected. Assume that every week all 100 tickets are sold and we start afresh the next week with a new lottery. If each ticket costs €3 and you buy 2 tickets every week, how much would you expect to pay overall on tickets by the time you win the lottery for the first time? \n" ); document.write( "

Algebra.Com's Answer #835217 by ikleyn(52814) $\"\"$ $\"About$

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\n" ); document.write( "A local club sells 100 tickets every week from which a winning ticket is randomly selected.
\n" ); document.write( "Assume that every week all 100 tickets are sold and we start afresh the next week with a new lottery.
\n" ); document.write( "If each ticket costs €3 and you buy 2 tickets every week, how much would you expect to pay overall
\n" ); document.write( "on tickets by the time you win the lottery for the first time?
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document.write( "From week to week, at each lottery, we have a binomial experiment \r\n" );
document.write( "with the probability of the success (winning at each new week/lottery) of \r\n" );
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document.write( "    p = P(1st ticket wins) + P(1st ticket does not win, but the 2nd ticket wins) = \r\n" );
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document.write( "      =  +  =  =  =  = 0.02.\r\n" );
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document.write( "Repeating this experiment every week (formally, infinitely many times) and paying 2*3 = 6 euros \r\n" );
document.write( "every week for 2 tickets, we get the mathematical expectation of the amount of the money \r\n" );
document.write( "we pay overall on tickets by the time you win the lottery for the first time\r\n" );
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document.write( "    E = 6p + 12(1-p)*p + 18(1-p)^2*p + 24(1-p)^3*p + 30(1-p)^4)*p + . . . =\r\n" );
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document.write( "      = 6p*(1 + 2(1-p) + 3(1-p)^2 + 4(1-p)^3 + 5(1-p)^4 + . . . ).\r\n" );
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document.write( "The formula of the infinite sum in the brackets is classic\r\n" );
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document.write( "        1 + 2*x + 3x^2 + 4x^3 + 5x^4 + . . . = .   (*)\r\n" );
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document.write( "You may find its proof everywhere (see the relevant info at the end of my post).\r\n" );
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document.write( "In our case, x = 1-p, and we get\r\n" );
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document.write( "    E =  =  =  =  = 6*50 = 300 euros.\r\n" );
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document.write( "ANSWER.  You expect to pay 300 euros overall on tickets by the time you win the lottery for the first time.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Below is the link to the formula (*), first one, which I found in the Internet.\r
\n" ); document.write( "\n" ); document.write( "https://www.toppr.com/ask/en-us/question/find-the-sum-of-the-series-1-2x-3x2-4x3-dotsgiven-that/\r
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\n" ); document.write( "\n" ); document.write( "You may find many sources for it, and, without any doubts, you may find it
\n" ); document.write( "in any serious textbook on combinatorics - it is like a PREREQUISITE for solving this problem.\r
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