Suppose each box of a popular brand of cereal contains a pen as a prize. The pens come in four colors: blue, red, green, and yellow. Each color of pen is equally likely to appear in any box of cereal,
What is the probability of having to buy at least five boxes of cereal to get a blue pen? Show all of the work that leads to your solution.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4

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