Not going to lie, this question haunted me for a couple days as it has a lot to juggle in terms of restrictions and extra information. I apologize for the delay between now and my poorer reply before.

So the best thing I've come up with reference to the question is:

1.**"In a species of beetle, red body color is dominant to brown. Two red beetles are crossed and produce 31 red and 9 brown offspring (F1 generation).**

Your parent genotypes: Rr and Rr

F1 genotypes : RR, Rr, Rr and rr.

2.**If two red F1 beetles are crossed, what is the probability that both red and brown beetles will appear in the F2 generation? (Note: Assume Mendelian inheritance patterns.)"**

**If you cross two red F1 beetles, you are only considering the 31 red offspring which should be roughly 2/3 Rr and 1/3 RR. This is how you get the 3 in the denominator by eliminating the possibility of using rr artificially.**

3.From here, you can either have an RR x Rr cross or an Rr x Rr cross.

*What you want is the chance that you will get brown and red offspring, which is any cross that is not involving RR as one of the parents.*

As a result, I can calculate the probability of red and brown offspring (PRB) by the following equation:

PRB = 1 - PR

Where PR is the probability of red only offspring.

PR can be described as 2 different crosses:

RR x Rr which has a probability of (1/3)*(2/3) = 2/9

and RR x RR which has a probability of (1/3)*(1/3) = 1/9

PR is the sum of these two independant possibilities, so we get 3/9 = 1/3 probability of red only offspring.

Therefore PRB = 1 - (1/3) = 2/3.

Needless to say, this took me a couple of days to chew on and come up with a somewhat reasonable answer. I definitely wouldn't have gotten this on the MCAT. I would say it's a question I would have flagged and guessed, and likely gotten wrong.

Please let me know if this is still unclear, I hope this helps!

Katt