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In a lottery, a ticket costs $\$3$, and the jackpot is worth $\$250,000$. In all, $100,000$ tickets are sold, one ticket is randomly drawn, and the winner is awarded the jackpot. What are the expected winnings in dollars of buying one ticket (including the cost of the ticket)?

Hello! I am a middle school student, so simply worded answers would be very appreciated. I know there is a $\frac{1}{100000}$ chance of winning the jackpot, so I multiplied that by $250000 - 3$, since I think that is the value of the winnings if you first buy a ticket, which costs $3.

How do I proceed from here? Am I meant to subtract by $3 \cdot \frac{99999}{100000}$, since you end up losing $3?

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  • $\begingroup$ No, your average gross return is \$2.50 so the average net return is \$2.50-\$3.00=$-50$ cents.. You don't take the cost of the ticket into account until the end. Another way of looking at it is that the lottery promoters collected \$300,000 and paid out \$250,000, so they cleared \$50,000. So each purchaser kicked in 50 cents to whatever the cause is. $\endgroup$ – saulspatz Apr 15 '18 at 18:36
  • $\begingroup$ $\left(\frac{1}{100{,}000}\cdot\$250{,}000\right) - \$3 = \$2.50 - \$3 = -\$0.50$ is correct, and there is nothing more you need to do. You lose half a dollar on average. $\endgroup$ – Jeppe Stig Nielsen Apr 16 '18 at 20:39
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Phrased in the simplest terms possible, your expected value is what you stand to gain times the likelihood of gaining it plus what you stand to lose times the likelihood of losing it. In formulas $E(X) = (250000 - 3) \cdot \frac{1}{100000} - 3 \cdot \frac{99999}{100000} = 0.5 $

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