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This problem is from Problem Solving Strategies - Crossing the River with Dogs and Other Mathematical Adventures by Ken Johnson and Ted Herr.

I first let capital letters denote shirts, and lowercase letters denote ties (with one letter representing each to make things easier).

So, B = blue, G = green, W = white, P = blue and green print, and S = red and white striped shirt.
I also made w = white, b = blue, y = yellow, p = green and blue striped tie

Then, I began working out the possible combinations:

For (color) shirt:

Blue (no blue tie, pattern much include blue [as denoted in the problem]): Bw, By, Bp

Green (no green tie, patter much include green): Gw, Gb, Gy, Gp

White: Wb, Wy

Blue and Green Pattern (P): Pb

Red and white stripe (S): Sw

I am wondering if there are any possibilities I missed. As of now, I have worked out there to be 11 combinations, with 6 out of the 11 involving the color blue. Any help would be much appreciated.

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    Ah, that is the one! I simply glossed over it since I saw green immediately. Thanks.2012-03-03

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One relatively easy systematic approach is to make a table for the combinations; here I’ve listed the shirts down the side and the ties across the top. I’ve put a $b$ in the cells corresponding to acceptable combinations that include blue, and a $y$ in the cells corresponding to other acceptable combinations.

$\begin{array}{r|c} &\text{white}&\text{blue}&\text{yellow}&\text{green/blue}\\ \hline \text{blue}&b&&b&b\\ \hline \text{green}&y&b&y&b\\ \hline \text{white}&&b&y\\ \hline \text{blue/green}&&b\\ \hline \text{red/white}&y \end{array}$

As you can see, there are indeed $11$ acceptable combinations, but seven contain blue. Thus, if he is equally likely to pick any of the acceptable combinations, his probability of picking one that includes blue is $\frac7{11}$.

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    Ah. Clever solution with the table - nice way to organize the information! Thanks.2012-03-03