Q.4.31 - A First Course in Probability

Question

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability $p$, then he or she will receive a score of

$1-(1-p{)}^2\text{ if it does rain }$

$1-p^2 if it does not rain$

We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability $p^{*}$, what value of $p$ should he or she assert so as to maximize the expected score?

Step-by-Step Solution

Answer

The value that he/she should get as $p=p^{*}$to the maximum of the wished score.

Step 1: Given information

To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability $p$, then he or she will receive a score of

$1-(1-p{)}^2\text{ if it does rain }$

$1-p^2\text{ if it does not rain }$

We need to find that value of $p$ should he or she assert so as to maximize the expected score

Step 2: Solution

If we are calculating the expression to $0$ so we can find that the value is,

$2p^{\ast }-2p=0$

$\Rightarrow 2p^{\ast }=2p$

$\Rightarrow p=p^{\ast }$

So for $p=p^{*}$, the predicted score of the meteorologist will be maximum So if an individual actually thinks that it will rain tomorrow with possibility ${\mathrm{p}}^{*}$, Then

He/she should claim with $p=p^{*}$ to maximize the wished score.

Step 3: Final answer

From the calculation it is clear that the value that he/she should get as $p=p^{*}$to the maximum of the wished score