probability - Blood Type Puzzle 1: Am I Pure Blood?

If my parents (biological) and I all have blood type B, how likely am I a pure blood? (expressed as a percentage)

Follow up question: If my dad is blood type AB instead of blood type B, and everything else is the same as above, will this change increase or decrease my chance of being a pure blood? (can you answer this without extensive calculation?)

Explanation and Clarification:

• The blood type in this question refers to the ABO blood type controlled by the A, B, O genes. Blood type A is either AA or AO, B is BB or BO, O is OO and AB is AB.


• A pure blood in this question refers to a person having two identical blood type alleles, that is all people of blood type O (OO), some people of blood type A and B (AA, BB) but no one of blood type AB.



• Unless specified otherwise, a blood type A or blood type B person has a 50% chance of being a pure blood (AA, BB)

Answer

My parents' blood types can be one of four scenarios, which are equally-likely since they are not conditioned on my own blood type (clarified in comments):

In MMFF format:
1. BBBB - 25% chance
2. BBBO - 25% chance
3: BOBB - 25% chance
4: BOBO - 25% chance

How likely is it for me to have inherited the different possibilities, under each of these parental scenarios?

1. Parents BBBB -> Me: 100% BB
2. Parents BBBO -> Me: 50% BB, 50% BO
3. Parents BOBB -> Me: 50% BB, 50% BO
4. Parents BOBO -> Me: 25% BB, 50% BO, 25% OO

Multiplying by 4 to turn these into integers instead of percentages, we get the following possibilities:

I am BB: 9 occurrences
I am BO: 6 occurrences
I am OO: 1 occurrence

But we have prior knowledge!

I know I am blood type B, meaning we need to exclude the possibility of OO. That means there are 15 possibilities, 9 of which are BB.

Therefore, the probability that I am pureblood is

9/15 = 0.6 (60%)

But what if my father is type AB? Without recalculating everything, we can quickly tell that given knowledge that I am type B,

We know for sure I got the B from my father. So the only relevant question is, what did I get a B or an O from my mother? Since she can be BB or BO, with equal probability, 3/4 = 75% of the time I will be pureblood.

Or, less math-y, since an O was replaced by an A, there are fewer chances for an O to dilute the purity of my B. (Kind of a funny way to think about purity...)

(NB: I realize these are the same numbers as @JMP's earlier answers, but I wanted to provide a more extensive explanation)