Question

A lumber mill produces boards that are 2,4,6,8, or 10 meters long. They are given trees with lengths uniformly distributed between 0 and 12 meters long and make one board from each tree, using as much wood as possible from each tree. Determine, approximately, the probability that after1000 trees are harvested, the length of wasted wood exceeds x meters. Simplify your answer.

Answer 1

the length of wood wasted will be uniformly distributed in [0,2]

because in the whole range 0 to 12 if we get tree of any length we can produce boards of length of any even number between 0 and 12 so the lrngth wood wasted can be the length between two successive even number which is 0 to 2

{we can't waste more than 2 metres because if lets say wastage = 2.5 then make one board of 2 metre and remaining wood is 0.5}

for 1000 trees waastage will be between 0 and 2*1000 metres uniformly distributed

P(x>=X) = (2000-X) / (2000-0) {x between 0 and 2000}

P(x>=X)= 1 - X/2000 {if X <=2000}

= 0 {if X > 2000}

(please UPVOTE)