# Raffles, Deworming, and Statistics

Sometimes statistics can help when it’s hard to decide what to do.

You’re at a local art fair, and they’re raffling off a car worth $10,000. Five hundred tickets are being sold, each for$10. Does it make financial sense to buy a ticket? (For the moment, let’s set aside other questions about raffles and just focus on the benefit for you, the potential ticket-buyer.)

You can use a statistical concept called “expected value” to help you decide. Expected value is calculated by multiplying the probability of each potential outcome by its value, then adding these results together to get the average result of an action.

Let’s figure this out—a car is on the line. First, we multiply the probability of each potential outcome by its value.

We might win the car. Assuming all tickets are sold, the probability of winning the car is 1 in 500, and the value of winning the car is $10,000. 1/500 x$10,000 = $20 We probably won’t win the car. The probability of not winning the car is 499 in 500. 499/500 x$0 = $0 Then, we add the results together to get the average result of an action.$20 + $0 =$20

Thus, the expected value of purchasing a raffle ticket under the conditions specified above is $20. That$20 represents the average result of buying a raffle ticket. It’s twice the ticket’s cost, making the raffle ticket a pretty good bet. In any one instance, you probably won’t win, but if you repeatedly make these kinds of bets, over time you’re likely to come out ahead.

(One important note that we’ll come back to later: Because it’s an average, the expected value doesn’t indicate what we think will actually happen in any specific instance. In fact, in this case our action of buying a raffle ticket cannot generate the expected value. We cannot win $20. We can win the$10,000 car or nothing.)

You might be saying to yourself: I’m still not that convinced about buying a raffle ticket, but I’m even less sure how this relates to GiveWell’s funding decisions.

GiveWell sorts through hundreds of funding opportunities, looking for the ones that are most cost-effective. We decide among multiple programs that differ from one another, not just in terms of the conditions they treat, the interventions themselves, or the locations they serve, but also in terms of the information we have about them and what we’re most uncertain about. We create models that incorporate a range of factors to estimate cost-effectiveness and decide which programs we ought to fund. Expected value is an important tool in those models.

## One example of expected value in practice: Deworming

Hundreds of millions of people have parasitic worm infections. We know that inexpensive medications can effectively kill parasitic worms. However, we have much more limited evidence on the long-term impact from clearing worm infections (as we’ve written about here and here).

Some studies indicate that reducing worm infections during childhood can have a significant later impact on income during adulthood. Other studies indicate that deworming has a negligible effect. Thus, we calculate the expected value of those possibilities (a small chance of a large effect, and a much larger chance of almost no effect) in order to determine the program’s cost-effectiveness. The outcome is analogous to our car raffle: because deworming is quite inexpensive, the program’s expected value often meets our cost-effectiveness threshold, as in this recent grant, even though we believe that most likely the program has very little if any impact on livelihoods.

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Why does this matter? Understanding expected value calculations means that you can’t always interpret our cost effectiveness estimates in the way you might think intuitively. The average value allows us to compare programs against one another, but it doesn’t necessarily indicate a value that can actually occur, and sometimes the average value is the result of (say) a large probability of no effect combined with a small probability of an exceptionally large effect.

In addition, two programs might have similar expected values but might differ in the range of possible outcomes. We account for this in two of our Giving Funds. Our Top Charities Fund only includes programs where the expected value is high and we are highly confident in the impact—that is, there is a high probability of a highly positive impact. In contrast, the All Grants Fund supports some grants with higher expected value but also a higher risk of not achieving any impact whatsoever. This way, donors can choose to support the fund that aligns with their preferences.

### Comments

• Sebastian on August 22, 2024 at 2:07 am said:

My intuition is that I could double my investment if I were able to by all tickets:

500 x $10 =$5,000
$10,000 –$5,000 = \$5,000

And therefore it should also be a good deal to buy less than all tickets. But I’m not a mathematician so I wonder if that’s another way to solve the problem, the same way in another disguise or simply a false assumption.