QUESTIONS AND PROBLEMS 531hiding itself very well. The curious reader is referred to Problem 18.7, where the
paradox is explained with an even more extreme example.Questions and problems
18.1. Weather forecasters are evaluated for accuracy using the Briers score. The
a posteriori probability of rain on a given day, judged from the observation of that
day, is 0 if rain did not fall and 1 if rain did fall. A weather forecaster who said
(the day before) that the chance of rain was 30% gets a Briers score of 30^2 = 900
if no rain fell and 70^2 = 4900 if rain fell. Imagine a very good forecaster, who over
many years of observation learns that a certain weather pattern will bring rain 30%
of the time. Also assume that for the sake of negotiating a contract that forecaster
wishes to optimize (minimize) his or her Briers score. Should that forecaster state
truthfully that the probability of rain is 30%? If we assume that the prediction and
the outcome are independent events, we find that, for the days on which the true
probability of rain is 30% the forecaster who makes a prediction of a 30% probability
would in the long run average a Briers score of 0.3- 70^2 + 0.7-30^2 = 2100. This score
is better (in the sense of a golf score—it is lower) than would result from randomly
predicting a 100% probability of rain 30%> of the time and a 0%> probability 70% of
the time. That strategy will be correct an expected 58% of the time (.58 = .3^2 + .7^2 )
and incorrect 42% of the time, resulting in a Briers score of .42 · 100^2 = 4200. Let
ñ be the actual probability of rain and ÷ the forecast probability. Assuming that
the event and the forecast are independent, show that the expected Briers score
104 (p(l — x)^2 + (1 — p)x^2 ) is minimized when ÷ = p. [Note: If this result did not
hold, a meteorologist who prized his/her reputation as a forecaster, based on the
Briers measure, would be well advised to predict an incorrect probability, so as to
get a better score for accuracy!]
18.2. We saw above that Cardano (probably) and Pascal and Leibniz (certainly)
miscalculated some elementary probabilities. As an illustration of the counter-
intuitive nature of many simple probabilities, consider the following hypothetical
games. (A casino could probably be persuaded to open such games if there was
enough public interest in them.) In game 1 the dealer lays down two randomly-
chosen cards from a deck on the table and turns one face up. If that card is not an
ace, no game is played. The cards are replaced in the deck, the deck is shuffled, and
the game begins again. If the card is an ace, players are invited to bet against a
fixed winning amount offered by the house that the other card is also an ace. What
winning should the house offer (in order to break even in the long run) if players
pay one dollar per bet?
In game 2 the rules are the same, except that the game is played only when the
card turned up is the ace of hearts. What winning should the house offer in order
to break even charging one dollar to bet? Why is this amount not the same as for
game 1?
18.3. Use the Maclaurin series for e-'^1 /^2 )'^2 to verify that the series given by de
Moivre, which was