Of three prisoners, one has been chosen by lot for execution. Call the prisoners $1,2,$ and $3.$ Prisoner $3$ says to the guard: $$Which of $1$ and $2$ is not to be executed? One of them won$$t be$,$ and you reveal no information about myself by telling me which one will not be executed.$$ The guard finds this reasonable, and replies $$Prisoner $1$ is not to be executed.$$

Suppose that if the guard has a choice in which of prisoners $1$ and $2$ to name $($that is$,$ if $3$ is to be executed$)$ the guard chooses to name $1$ with probability

$.$ If

$,$ what is the conditional probability that prisoner $3$ will be executed given the information provided by the guard?