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Objective Versus Subjective Probability

Before any very large lottery drawing there will be quotes from prospective players usually taking an optimistic tone .They’ll say things like “you can’t win if you don’t play”, “somebody will win and might as well be me”, and so on. Of course the odds of a single ticket winning the largest most popular US lotteries are on the order one in 175 million.

Overestimation of probabilities

It’s hard to think about such long odds of winning, the numbers are so large that they are hard to make tangible without comparison to something else. So let’s think about these odds in terms of time. The mathematics used to calculate the probability of lotteries like this along with the idea of an expected value of a decision were formally described in 1654 by the mathematician Blaise Pascal.

Believe it or not, the probability of a single ticket winning a mega millions lottery is almost identical to the probability of randomly choosing one single second from all the years days hours and minutes since 1654 and then picking the very second that you were born. But, does overestimation of probabilities explain the popularity of lotteries.

Reward anticipation

Not completely. There are at least two other factors that contribute. One factor comes from the way our brains process rewards. As discussed in an earlier post the brain’s dopamine system doesn’t simply respond to rewards themselves. It responds to signals about future rewards which can lead to anticipation of what could occur.

The sense of possibility the fantasy about what we can do with their wings can be very motivating and it isn’t necessarily irrational to spend one dollar on a ticket in order to have those feelings of anticipation. Also important are the social aspects of playing the lottery. People play in groups and discuss the upcoming jackpots with their neighbors in those social factors can be powerful motivators for playing.

Might events

Let’s move from these low probability events to events of intermediate probability for about 25% to 75%. I’ll refer to these as “might events” they might happen or they might not. Within the might range the probability weighting function flattens out, meaning that in a change in objective probability doesn’t have that much of subjective effect.

Let me illustrate this with the thought problem. Suppose that you go to your primary care physician and you are told that because of your genetic profile you have a 5% probability of contracting a rare form of cancer in the next five years. There is a drug treatment that is completely effective, it’s expensive and it has side effects, but it will eliminate any chance you will develop this cancer.

It will take your probability of cancer from 5% down to 0%. Do you begin that drug treatment? When faced with these sorts of scenarios people tend to be willing to undergo treatment, going from 5% to 0% chance of cancer seems pretty dramatic.

Objective versus subjective probability

Now let’s modify this thought problem suppose that your genetics indicator you actually have a 60% probability of getting cancer, but at the same drug treatment could reduce your probability down to 55%. You’ll incur the same expense and face the same side effects for a change from 60% to 55%. Do you begin that drug treatment?

This situation is much harder for most people. The difference between 60% and 55% seems pretty minor. that doesn’t seem like the sort of change shall be worth taking on an expensive and painful course of treatment. People might say I might get cancer or I might not regardless so I don’t need to ruin my quality of life in the meantime.

As a rough guide within this middle range for every 2% change in objective probability there is slightly less than a 1% change in subjective probability.