Perceived risks and benefits

People are sometimes horribly bad at reasoning about probabilities. cakmpls has just written about this, focusing on the issue of perceived risk. Her main example: women are ten times more likely to die of heart disease than breast cancer, and yet they consistently report being more frightened of the latter. She has several good theories about why this should be, and her commenters supply more. I wanted to mention the other side of the coin, the perception of potential benefits, and it seemed I was going a little too off-topic, so I'm not leaving it as a comment.

The probability that something will happen is often written as a fraction between 0 and 1, where 0 is "impossible" and 1 is "certain". To find the expected number of occurrences of the event in a given number of trials, you multiply the number of trials by the probability. For example, if there's a probability of 0.103 of winning a particular solitaire game, and you play 3000 times, you'll win about 309 times (3000 x 0.103).

To find out what the event is "worth", you multiply the probability by the value of the event occurring. Suppose there's a game where you have one chance in a hundred of winning a thousand dollars. The carny says you can play for $5. Should you? The "right" answer is yes, because $1000 x 0.01 is $10, so playing the game is worth $10; you're getting a $10 value for a $5 outlay.

It seems that we are really bad at deciding what to do when the probabilities are very small and the risks or benefits are very large. Suppose it costs a dollar to play the lottery, and you have one chance in a hundred million at winning ten million dollars. Rationally, a play is worth about a dime; but it turns out in real life that people will line up down the block for a chance to play. This is good for the people who run the lottery, because they will collect a hundred million bucks for every time they have to pay out ten million.

The overall pattern seems to be: when the probability is tiny and the risk or benefit is very large, our intuitions are rotten, and we tend to act as if the probability were the smallest we can really understand (somewhere between 0.01 and 0.001, between one in a hundred and one in a thousand). The result is that the risk or benefit is drastically exaggerated.

This might explain the breast cancer fear as well. The chances of breast cancer or heart disease are well under our probability "radar", so we treat them as essentially equal.