# 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)
• To further clarify, a child with blood type A no longer has a 50% chance of being pure blood if we have information regarding their parent or parents' blood type. But not the other way. – Manto Nov 24 '19 at 11:44
• this means, the parents have a 50% of being pure in this question while the child's blood type is based on the parents – Manto Nov 24 '19 at 12:14

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)

• I like your explanation the best. – JS1 Nov 24 '19 at 23:13

According to:

it is:

$$60\%$$ likely that you are pure blood. For both you and both parents to be B, there are $$15$$ possibilities, of which $$9$$ are BB.

Q2:

It will increase the chance that you are a pure blood (to $$75\%$$), because of the $$8$$ possibilities, only $$4$$ are type B, of which $$3$$ are BB.

• not correct unfortunately. read the question again :) – Manto Nov 24 '19 at 11:22
• @Manto; is it right now? – JMP Nov 24 '19 at 11:30
• you are getting close. Your edit was in the right direction for sure. But there is still one thing off I believe. – Manto Nov 24 '19 at 11:35
• @Manto; got it? – JMP Nov 24 '19 at 11:37
• just to clarify a child with blood type A no longer has a 50% chance of being pure blood if we have information regarding their parent or parents' blood type. But not the other way. Also, can you answer the followup question, do you think I am more likely or less likely to be pure blood if my dad is AB instead of B? – Manto Nov 24 '19 at 11:45

The answer is not 60% it is

58.33333333%

The parents have a 25% chance being BBBB, 50% chance being BBBO and 25% chance being BOBO. This should not change when eliminating OO.

In the case of BBBB, the child is BB for sure, which is 100% chance. In the case of BBBO, where one parent is BB and the other is BO, the child is 50% BB depending on whether the BO parent gives them B or O. In the case of BOBO, 25% chance the child is BB, 50% the child is BO and 25% chance is OO.

But because we know the child is not OO, we need to eliminate the OO from BOBO, but BOBO is still 25%, while BB is now 1/3 and BO is 2/3.

25% * 100% + 50% * 50% + 25% * 1/3 = 58.333333%

If the father is AB, the chance being BB will

Increase

because if the parents are both AB, and the child is B, then the child is 100% BB. So having AB parent will increase the chance because they can only give the "B" from AB

• thanks. I was trying to imply that if both parents being AB gives you BB for sure, either parent being an AB instead of B, would thus increases the chance. – David Nov 24 '19 at 23:04
• Actually on further thought I think the 58.3% should be 60%, because the problem with your calculation is that the prior knowledge that the child is not OO changes the probability that the parents are BBBB or BBBO (i.e. the chances are no longer 25% any more). Similar to the Monty Hall problem. – JS1 Nov 24 '19 at 23:06