Quotes About Probability

People who buy lottery tickets in vast amounts show themselves willing to pay much more than expected value for very small chances to win a large prize.

When the top prize is very large, ticket buyers appear indifferent to the fact that their chance of winning is minuscule. A lottery ticket is the ultimate example of the possibility effect. Without a ticket you cannot win, with a ticket you have a chance, and whether the chance is tiny or merely small matters little. Of course, what people acquire with a ticket is more than a chance to win; it is the right to dream pleasantly of winning.

the weighted average of its possible dollar outcomes; it is the average of the utilities of these outcomes, each weighted by its probability.

Statistics produce many observations that appear to beg for causal explanations but do not lend themselves to such explanations. Many facts of the world are due to chance, including accidents of sampling. Causal explanations of chance events are inevitably wrong.

Mutual funds are run by highly experienced and hardworking professionals who buy and sell stocks to achieve the best possible results for their clients. Nevertheless, the evidence from more than fifty years of research is conclusive: for a large majority of fund managers, the selection of stocks is more like rolling dice than like playing poker. Typically at least two out of every three mutual funds underperform the overall market in any given year.

extreme outcomes (both high and low) are more likely to be found in small than in large samples. This explanation is not causal.

An unbiased appreciation of uncertainty is a cornerstone of rationality

risk-averse decision maker will choose a sure thing that is less than expected value, in effect paying a premium to avoid the uncertainty.

For example, if you believe that 3% of graduate students are enrolled in computer science (the base rate), and you also believe that the description of Tom W is 4 times more likely for a graduate student in that field than in other fields, then Bayes's rule says you must believe that the probability that Tom W is a computer scientist is now 11%. If the base rate had been 80%, the new degree of belief would be 94.1%.

we are statistically punished for being nice

The hot hand is entirely in the eye of the beholders, who are consistently too quick to perceive order and causality in randomness.

Both in explaining the past and in predicting the future, we focus on the causal role of skill and neglect the role of luck. We are therefore prone to an illusion of control.

We are prone to overestimate how much we understand about the world and to underestimate the role of chance in events. Overconfidence is fed by the illusory certainty of hindsight.

If an event had actually occurred, people exaggerated the probability that they had assigned to it earlier.

We are prone to overestimate how much we understand about the world and to underestimate the role of chance in events. Overconfidence

There is one thing you can do when you have doubts about the quality of the evidence: let your judgments of probability stay close to the base rate. Don't expect this exercise of discipline to be easy—it requires a significant effort of self-monitoring and self-control.

for a large majority of fund managers, the selection of stocks is more like rolling dice than like playing poker. Typically at least two out of every three mutual funds underperform the overall market in any given year.

The comparison of firms that have been more or less successful is to a significant extent a comparison between firms that have been more or less lucky.

The general tendency to overestimate the probability of conjunctive events leads to unwarranted optimism in the evaluation of the likelihood that a plan will succeed or that a project will be completed on time.

I smiled and said nothing. But I thought, "Well, I took it away from you this morning. If your success was due mostly to chance, how much credit are you entitled to take for it?

If an event that was assigned a probability of 90% fails to happen, the judgment of probability was not necessarily a bad one. After all, outcomes that are just 10% likely to happen end up happening 10% of the time. The Gambardi exercise is an example of a nonverifiable predictive judgment, for two separate reasons: Gambardi is fictitious and the answer is probabilistic.

judgments of similarity and probability are not constrained by the same logical rules. It is entirely acceptable for judgments of similarity to be unaffected by base rates and also by the possibility that the description was inaccurate, but anyone who ignores base rates and the quality of evidence in probability assessments will certainly make mistakes.

Relying on causal thinking about a single case is a source of predictable errors. Taking the statistical view, which we will also call the outside view, is a way to avoid these errors.

Our predilection for causal thinking exposes us to serious mistakes in evaluating the randomness of truly random events.