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Given in the question that, a prisoner is trapped in a cell containing$3$doors. The first door leads to a tunnel that returns him to his cell after $2$days鈥� travel. The second leads to a tunnel that returns him to his cell after $4$days鈥� travel. The third door leads to freedom after $1$day of travel. If it is assumed that the prisoner will always select doors$1,2,$and $3$with respective probabilities $.5,.3,$and .$2,$what is the expected number of days until the prisoner reaches freedom?

Allow $X$to signify the quantity of days until the detainee get away and $Y$ mean the entryway the detainee picks. Then

Now, $E[X鈭�Y=1]=E\left[X\right]+2$, since the prisoner will essentially get back to the cell and the issue begins once again.

Similarly, $E[X鈭�Y=2]=E\left[X\right]+4.E[X鈭�Y=3]=1$, of course. Hence,

$E\left[X\right]=.5\left(E\right[X]+2)+.3\left(E\right[X]+4)+.2=.5路E\left[X\right]+1+.3路E\left[X\right]+1.2+.2=.8路E\left[X\right]+2.4$

$\mathbb{E}\left[X\right]=12$.

The expected number of days until the prisoner reaches freedom is

$\mathbb{E}\left[X\right]=12$.