A 6 An educational software company produces two software packages: an
algebraic solver and a graphing program. They project a demand for at least
100 algebraic solvers and 80 graphing programs each day. They can produce
up to 200 algebraic solvers and 170 graphing programs per day. They need
to produce at least 200 software packages each day to satisfy existing orders.
Each algebraic solver makes a loss of $1.50. Each graphing program makes
a profit of $4.50.
Determine how many of each software package the company should
produce each day to maximize its profits.

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.