If 63% of people think something will happen, what is the probability people give to its happening?

The mischievous Lord Norton of Louth has written a blog post about the three ways in which Donald Trump might leave office. As Norton notes,

Donald Trump is remarkable for the extent to which his possible resignation or removal is being discussed so early in his incumbency

Removal by impeachment is sufficiently plausible that we have some surveys asking people how likely they think it is. According to two such surveys, a majority of American professors (63%) and a near-majority amongst the general population (48%) think that Trump will be impeached.

Both of these numbers seemed to me to be very high indeed, but as @JoshuaBBlake noted, the proportion of people who believe that something will happen can lead to very different judgments as the probability of that thing:

More generally, if x% of people think that some event will happen, what do people in general think the probability of that event happening is?

Ideally, we'd have additional data that would help us answer this question. But where we only have information on proportions of respondents, we can produce a reasonable estimate by averaging lower and upper bounds.

We can produce a lower bound by supposing that

  • the x% of people who think the event will happen assign it the lowest possible probability consistent with the event being more likely than not (50% + some small constant), and that
  • the (1-x)% who think the event won't happen assign it zero probability.

The lower bound then becomes slightly greater than 50% of x, or one half of x.

We can produce an upper bound by supposing that

  • the x% of people who think the event will happen assign it the highest possible probability (100%), and that
  • the (1-x)% who think the event won’t happen assign it the highest possible probability consistent with the event being less likely than not (50% minus some small constant).

The upper bound then becomes x + (1-x)/2. The probability people give to this event must be somewhere between the upper and lower bound, and if we have no further information it's reasonable to assume it's equidistant between the two.

After a bit of re-arranging, this simplifies to 1/4 plus 1/4 of the proportion who think it will happen. This means that:

  • If the proportion who think Trump will be impeached is 63%, then the probability people attach to Trump being impeached ought to be around 56.5%.
  • If the proportion who think Trump will be impeached is 48%, then the probability people attach to Trump being impeached ought to be around 49%

This is shown visually in the following graph, where the black line indicates the average of the upper and lower bound (indicated by the shaded region):

You might think that the true shape of this line, for most issues, is slightly different. A common choice might be a lazy-S, or sigmoid curve which is steepest at 50% — but I'm not sure how one could sensibly derive this from the kind of minimal information represented by upper or lower bounds.

I'm not aware of any research on the shape of this curve — probably because there's been a huge amount of research on going more directly to people's (elicited) judgements about probability — so if you do know something, drop a comment below.

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