every time i use a piece of scrap paper, i crumple it up and try to shoot it inside the recycling bin across the room. i'm pretty good at it: if i shoot 5 pieces of paper at the recycling bin, at least one of them will make it inside the recycling bin with probability 211/243. if i shoot 6 pieces of paper at the recycling bin, what's the probability at least two of them make it inside the recycling bin?(assume that the probability i make any given shot is uniform across every shot. leave your answer in exact form; there should be no decimals in your answer.)
The probablility that I can get a crumpled up piece of paper into the recycling bin is 211/243. So, if I shoot 6 pieces of paper, then the probablity I can get at least two into the recyccling bin is
\(\left( \frac{211}{243} \right)^6 + 6 \left( \frac{211}{243} \right)^5 \left( \frac{22}{243} \right) + 15 \left( \frac{211}{243} \right)^4 \left( \frac{22}{243} \right)^2 + 25 \left( \frac{211}{243} \right)^3 \left( \frac{22}{243} \right)^3 + 6 \left( \frac{211}{243} \right)^2 \left( \frac{22}{243} \right)^4,\)
because I get two into the recycling bin if I get two, three, four, five or six. This works out to about 77.91%.
