I don’t think a mean of the probabilities is the correct way to average; I think the logistic of the mean of the log odds (suggested by Douglas Knight) is better, and averages 25% and 40% to ~32%. Obviously that’s not far off, so in this case it’s a nitpick. It might be the best way of handing estimates from a group though; a weighted average would even work if one trusts different members of the group differently.
For the truly crazy (crazy as in wanting to go to lots of extra work for not much gain), I think we can subvert our mental facilities by asking ourselves for the probabilities in the absolute best and worse cases for each side; since we’re not very good at those estimations, treat them as a the 25th and 75th percentile, and construct a beta distribution that matches those parameters. This, however, is a huge pain, because not only do you need to find two parameters, but the CDF of the beta function is not terribly convenient. You’d then have a mean and a standard deviation, and if those seem way off base, you might want to revise your estimates.
I don’t think a mean of the probabilities is the correct way to average; I think the logistic of the mean of the log odds (suggested by Douglas Knight) is better, and averages 25% and 40% to ~32%. Obviously that’s not far off, so in this case it’s a nitpick. It might be the best way of handing estimates from a group though; a weighted average would even work if one trusts different members of the group differently.
For the truly crazy (crazy as in wanting to go to lots of extra work for not much gain), I think we can subvert our mental facilities by asking ourselves for the probabilities in the absolute best and worse cases for each side; since we’re not very good at those estimations, treat them as a the 25th and 75th percentile, and construct a beta distribution that matches those parameters. This, however, is a huge pain, because not only do you need to find two parameters, but the CDF of the beta function is not terribly convenient. You’d then have a mean and a standard deviation, and if those seem way off base, you might want to revise your estimates.