Let’s say it snows in Sherbrooke one fourth of the days during the winter. Imagine that it is winter time now and the local newspaper tries to predict whether or not it will snow tomorrow. 3/4 of snowy days and 3/5 of sunny days (no precipitation) are usually correctly predicted by the previous day’s paper. Today’s edition of the newspaper predicts snow tomorrow. What is the probability that this prediction is accurate?
Let E1 represent a snowy day and E2 represent a sunny day.
$\mathrm{P}(\mathrm{E} 1)=1 / 4 \mathrm{P}(\mathrm{E} 2)=3 / 4$
A: Correct Prediction $\mathrm{P}(\mathrm{A} / \mathrm{E} 1)=3 / 4 \mathrm{P}(\mathrm{A} / \mathrm{E} 2)=3 / 5$
The probability is given by $P\left(E_{1} / A\right)=\frac{P\left(E_{1}\right) P\left(A / E_{1}\right)}{P\left(E_{1}\right) P\left(A / E_{1}\right)+P\left(E_{2}\right) P\left(A / E_{2}\right)}$
$P\left(E_{1} / A\right)=\frac{0.25 \times 0.75}{0.25 \times 0.75+0.75 \times 0.6}$
$P\left(E_{1} / A\right)=0.294$
The probability that this prediction is accurate is $29.4 \%$.