Ok so you have a whole bunch of different genes at work here, which is going to make a lot of possibilities. I'll try to explain by breaking things down into the separate genes. Remember though, rather than the colour you see being inherited as one big chunk, it is like layers, where you start with a base colour (either red or black, or occassionally that weird 'brown'), then each type of modifier adds another layer on top.
Also, remember the basics that each animal has 2 copies of each gene, which can say the same thing (homozygous) or different things (heterozygous). When an animal is bred, they only pass on one of the genes - which one is 50/50 chances just like a coin toss - and so each animal has one of the two genes from each parent.
1. Base colour. (Note I'm going to leave out that "weird brown" because it's not all that common in our beef breeds and would just make life confusing here. So a cow is going to start off genetically with either a red or black base. Black is dominant which means either two (homozygous) or one (heterozygous) copy of the black gene will make the cow appear black. On the other hand, to appear red, they must have two copies of the red gene (homozygous recessive). The implications of this is that there is red animals can only pass on red genes, but black animals, if homozygous, can still pass on red genes and have red calves.
In your case, because the cow is smoky, we know she has one copy of the black gene, and the other copy is quite likely red coming from PB charolais (I believe most of them are red based under their white).
2. Dilution is what makes Chars white. It's only partially dominant, so that in a double dose, the animal is diluted all the way to white, while in a single dose, they are only partly diluted, to smoky or tan (smoky for the black based animals, tan for the red based ones). So your cow has a single copy of the dilution gene, which is obvious both by her visual colour and her PB Char parent. This means she will pass on the dilution 50% of the time, and 50% of the time not.
3. The rat tailed defect, as far as I know, requires the cooperation of different genes together. The animal must be black based, diluted (with either the char or simmental dilution), and also get the actual rat tailed gene. Something to that effect. The fact that you had full sibs with and without it illustrates that 50/50 chance element. Has the cow ever had a rat tailed calf before? If not, it's possible she just doesn't have the rat tailed gene, but you can't tell for sure until she has one, so still not a bad idea to plan wisely.
4. The roan gene in shorthorns. This gene is codominant which means that the heterozygous has a different look than the two homozygotes. So unlike in the chars, where white comes from a dilution, in shorthorns whites are actually just homozygous for roan. Solid reds are homozygous for "not roan" and roans have one copy of each gene. So when you breed a solid colour to a white shorthorn (such as Sin city) you get 100% roans, as they get the roan from the white parent and the non-roan from the solid parent.
So put that all together and geez what a soup. But I'll try to summarize the possibillities
Smoke cow X White shorthorn =
Blue Roan
Smoky roan (not sure if that's an official term, but that's what I'm saying
)
Red roan
Tan roan
Smoke cow X Red Shorthorn =
Black
Smoky
Red
Tan
Smoke Cow X Roan shorthorn =
All 8 given in above 2
About the rat tail....your best bet would be to stick to red based bulls, so that you maintain that 50% chance of a red based calf, which can't have the defect.
