The Million Dollar Draw

The Million Dollar Draw has just launched nationwide. Millions of scratch cards are sold in shops, each with a 1 in 1,000,000 chance of winning the $1,000,000 jackpot. For that amount of money, you and your friend are willing to get creative with the law.

Your friend quickly builds a scanner that can read through the scratch coating and detect a winning card underneath. It never misses a real winner. However, about 1% of losing cards have printing artefacts that look identical to a winning pattern under the scanner. These artefacts are baked into the card, so a card that reads positive will always read positive no matter how many times you scan it.

Armed with a scanner each, you both head to different shops to scan scratch cards. After hours of searching, you find a card that tests positive with the scanner. You buy it and bring it to your hideout, where your friend is waiting for you. His luck had failed him, and he didn’t find a single card that tested positive. After eyeing your scratch card, he offers you $200 for it.

You consider it for a moment. On one hand, a million dollars is a life-changing amount of money. On the other hand, he did invent the scanner. Should you take the deal?

Solution

Yes, take the deal.

Imagine there are 100,000,000 scratch cards, with 100 of them being winning cards. The scanner catches it (it never misses). But it also falsely flags 1% of the 99,999,900 losing cards, which is 999,999 cards. So in total, roughly $1+\mathrm{999,999}=\mathrm{1,000,099}$1+999,999=1,000,099 cards get flagged.

Of the 1,000,099 flagged cards, only 100 are real winning cards. The probability your card is the winner is $\frac{100}{\mathrm{1,000,099}}\approx 0.01\%$1,000,099100​≈0.01%.

The expected value of keeping the ticket is $\frac{100}{\mathrm{1,000,099}}\times \$\mathrm{1,000,000}\approx \$99.99$1,000,099100​×$1,000,000≈$99.99. Since your friend is offering $200, you should take the deal.