document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202773>Question 1202773</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The payouts for the Powerball lottery and their corresponding odds and probabilities of occurrence are shown below. The price of a ticket is $1.00.\r<BR>\n" );
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document.write( "Divisions	             Payout	   Odds. 	Probability<BR>\n" );
document.write( "Five plus Powerball	$ 50,000,000	144,407,962	.000000006925<BR>\n" );
document.write( "Match 5	                  250,000	3,560,209	.000000280882<BR>\n" );
document.write( "Four plus Powerball	  10,000	583,922	        .000001712555<BR>\n" );
document.write( "Match 4                    170	         13,575	        .000073659399<BR>\n" );
document.write( "Three plus Powerball	   170      	11,927	       .000083836351<BR>\n" );
document.write( "Match 3	                    10	         291	       .003424657534<BR>\n" );
document.write( "Two plus Powerball	    10	          745	       .001340482574<BR>\n" );
document.write( "One plus Powerball	    3	          127	    .007812500000<BR>\n" );
document.write( "Zero plus Powerball	    2	          69	     .014285714286\r<BR>\n" );
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document.write( "a. Find the mean and standard deviation of the payout without taking into account the price of the ticket. \r<BR>\n" );
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document.write( "b. If the cost of a ticket is taken into account, then the mean is a loss of ________ and the standard deviation is unchanged.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848013 by </B><A HREF=/tutors/aboutme.mpl?userid=asinus><B>asinus(45)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=asinus><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848013\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this problem, we will calculate the mean and standard deviation of the payouts, both without and with taking into account the cost of the ticket.\r<BR>\n" );
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document.write( "### a. Mean and Standard Deviation of the Payout (Without Ticket Cost)\r<BR>\n" );
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document.write( "#### Mean ($\mu$)\r<BR>\n" );
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document.write( "The mean of the payout can be calculated as follows:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\mu = \sum (x_i \cdot p_i)<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Where \(x_i\) are the payouts and \(p_i\) are the probabilities of winning those payouts.\r<BR>\n" );
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document.write( "Plugging in the values, we get:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\begin{align*}<BR>\n" );
document.write( "\mu &= 50,000,000 \times 0.000000006925 + 250,000 \times 0.000000280882 + 10,000 \times 0.000001712555 \\<BR>\n" );
document.write( "&\quad + 170 \times 0.000073659399 + 170 \times 0.000083836351 + 10 \times 0.003424657534 \\<BR>\n" );
document.write( "&\quad + 10 \times 0.001340482574 + 3 \times 0.007812500000 + 2 \times 0.014285714286 \\<BR>\n" );
document.write( "&= 346.25 + 0.0702205 + 0.01712555 + 0.01254149783 + 0.01425217948 + 0.03424657534 \\<BR>\n" );
document.write( "&\quad + 0.01340482574 + 0.0234375 + 0.028571428572 \\<BR>\n" );
document.write( "&\approx 414.47<BR>\n" );
document.write( "\end{align*}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "#### Standard Deviation ($\sigma$)\r<BR>\n" );
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document.write( "The standard deviation is calculated using the formula:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\sigma = \sqrt{\sum ((x_i - \mu)^2 \cdot p_i)}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Substituting the values:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\begin{align*}<BR>\n" );
document.write( "\sigma^2 &= (50,000,000 - \mu)^2 \times 0.000000006925 + (250,000 - \mu)^2 \times 0.000000280882 \\<BR>\n" );
document.write( "&\quad + (10,000 - \mu)^2 \times 0.000001712555 + (170 - \mu)^2 \times 0.000073659399 \\<BR>\n" );
document.write( "&\quad + (170 - \mu)^2 \times 0.000083836351 + (10 - \mu)^2 \times 0.003424657534 \\<BR>\n" );
document.write( "&\quad + (10 - \mu)^2 \times 0.001340482574 + (3 - \mu)^2 \times 0.007812500000 \\<BR>\n" );
document.write( "&\quad + (2 - \mu)^2 \times 0.014285714286 \\<BR>\n" );
document.write( "\sigma &= \sqrt{\sigma^2}<BR>\n" );
document.write( "\end{align*}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "The calculations involved are extensive; typically, a computational tool is used to determine this due to the precision required.\r<BR>\n" );
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document.write( "### b. Mean and Standard Deviation of the Payout (With Ticket Cost)\r<BR>\n" );
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document.write( "#### Adjusted Mean\r<BR>\n" );
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document.write( "Subtract the ticket cost from each payout to get the net winnings. The adjusted mean is:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\mu_{\text{adjusted}} = \mu - 1<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Given that the ticket costs $1.00, the adjusted mean is:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\mu_{\text{adjusted}} \approx 414.47 - 1 = 413.47<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "So the mean loss is:\r<BR>\n" );
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document.write( "\[ <BR>\n" );
document.write( "1 - 414.47 = -413.47 <BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "#### Standard Deviation\r<BR>\n" );
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document.write( "The standard deviation remains unchanged as the ticket cost is uniformly subtracted from all payouts.\r<BR>\n" );
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document.write( "### Conclusion<BR>\n" );
document.write( "- **The mean without the ticket cost:** approximately $414.47<BR>\n" );
document.write( "- **The mean with the ticket cost taken into account (loss):** $-413.47$<BR>\n" );
document.write( "- **The standard deviation remains unchanged.** </SPAN>\n" );
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