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I have very little background in probability and am trying to figure out how to determine the frequencies of the following combination of letters: YYRR, YYRr, YyRR, YyRr, yyRR, yyRr, YYrr, Yyrr, and yyrr. I know I can figure out those probabilities by making a chart below and counting the frequencies. But can someone explain how to compute the frequencies without making a chart?

Thanks in advance.

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    @Scaramouche: For instance, one might want to be able to do the same thing $f$or another case in which making a chart would be more work.2012-01-13

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There are four binary choices, y/Y and r/R for each of two parents, and all $2^4=16$ ordered combinations of these choices are being produced with equal probability. However, you want to count unordered combinations. Since there are three unordered combinations for each feature (YY, Yy, yy, and RR, Rr, rr, respectively) which can all be combined with each other, there are $3^2=9$ unordered combinations. The mixed combinations, Yy and Rr, correspond to two ordered combinations each, whereas the pure ones, YY, yy, RR and rr, correspond to one ordered combination each. Since it's the ordered combinations that are equiprobable, you get a factor of $2$ for each mixed feature. Thus the frequencies are:

YYRR: $1\cdot1=1$      YYRr: $1\cdot2=2$      YYrr: $1\cdot1=1$

YyRR: $2\cdot1=2$       YyRr: $2\cdot2=4$       Yyrr: $2\cdot1=2$

yyRR: $1\cdot1=1$        yyRr: $1\cdot2=2$       yyrr: $1\cdot1=1$