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.
Best Answer
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 $\frac7{11}$.