### Lotteries and Insurance: Safe Bets?

Brian
Slate's Tim Harford says you shouldn't buy rental car insurance unless you are a terrifically bad driver:

Indexed by tags economics, math, statistics, rental car, insurance, lottery, expected value, Deal or No Deal, Monty Hall Problem.

[F]or just $10 a day, I could protect myself from the frightening-sounding insurance deductible of $900—a sum I risked being charged if anything happened to the car. I bravely turned them down.Link. This is all fair enough, but, according to this golden oldie from Slate's Jordan Ellenberg, where you could really be making a smart investment is in the Powerball:

This was a strikingly overpriced offering. For each day's rental I was being asked to pay $10 to protect me from the risk of paying $900. The mathematics are hardly difficult: The insurance is fair only if I crash into something every 90 days. If I believed that, I wouldn't get behind the wheel at all.

. . . .

[A]nyone who pays even slightly more than the fair premium to escape from a risk on a $90 phone or a $900 insurance deductible must be making a mistake. The stakes are too tiny: In the context of a $1 million lifetime income, even $900 is a small enough risk to swallow. We should turn down these offers of insurance and save the money in a contingency fund to pay for the occasional loss. The odds would be well in our favor and the petty uncertainty shouldn't cause us a single sleepless night.

The question to ask is: What is the expected value of a lottery ticket? If the expected value is more than a dollar, and the ticket costs a dollar, you should buy a ticket. If the expected value is less than a dollar, you should keep your money.Let me interrupt to underscore this point, because if you understand it, Deal or No Deal is the stupidest, easiest, fish-in-a-barrel money-winning game show of all time. Since each case has the same odds of containing any given dollar value, all you have to do is add up all the money left on the board and divide by the number of cases left. That's the expected value of the eventual case you get to keep. If the deal Mr. Banker offers you is much less than that expected value, you'd be a fool to take it. If it's more, you'd be a fool not to. The rest of the show is theater and psychological tricks meant to draw you off your game. That's a hundred times easier than understanding the Monty Hall problem. Now back to our regularly scheduled article on the lottery:

So the masters of Powerball take in $200 million in ticket sales for Saturday's drawing. Very likely, they pay out $280 million in jackpot—not to mention the sub-jackpot prizes, which amounted to $41 million in Saturday's drawing. Which means the house loses. And if the house loses, by definition, the average player wins.Link. Well you don't have to tell me twice. Next time I won't make the same mistake as Jerry Seinfeld.

This may make Powerball look dumb; why would a casino run a game where the house stands to lose? The answer is that the current large jackpot is the result of a long string of games when the house did win. Cumulative-jackpot lotteries such as Powerball are essentially a massive transfer of value from the dupes who play when the jackpot is small to the wiser ones who wait until the jackpot is big, with the house taking a healthy cut along the way. Here's the one piece of solid advice in this column: If you play Powerball every day, stop playing Powerball every day. If your dollar can be spent for a 1 in 80 million chance of $10 million or a 1 in 80 million chance of $120 million, why would you choose the former?

Indexed by tags economics, math, statistics, rental car, insurance, lottery, expected value, Deal or No Deal, Monty Hall Problem.

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