Consider the Monty Hall problem:

Behind one door is a car; behind the others, goats. You pick a door, say No$.1,$ and the host

$($who knows what$$s behind the doors$)$ opens another door, say No$.3,$ which has a goat. He then

says to you, $$Do you want to pick door No$.2?$

Let Ai $=\{$car is behind door i$\}$ for i in $\{1,2,3\}.$ Before coming to the show you have studied

the data regarding the unknown car location, you have concluded that P $($A$1)=$ P $($A$3)=2$

$5$ and

P $($A$2)=1$

$5.$ Is it to your advantage to switch your choice? Use Bayes rule to justify your answer.

$1$