# Blood Type Puzzle 2: Am I Pure Blood?

This puzzle is similar to the first puzzle but with a few twists:

In this question:

• both my mom and I are blood type B.
• But this time I don't know my dad's blood type.
• My mom said her parents have the same blood type but is not the same as hers.

Given the above, how likely am I to be a pure blood? (Expressed as a percentage)

Explanation and clarification:

• 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.
• Pure blood 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) and if a person's blood type is unknown, we can assume he's equally likely to be A, B, AB or O unless affected by his or her parents or children

As JNF says, your mother must have AB parents and be BB herself.

Applying the final point directly to your father, before taking into account information from your own blood type your priors for his genotype are

OO: $$0.25$$, AB: $$0.25$$, AA: $$0.125$$, AO: $$0.125$$, BB: $$0.125$$, BO: $$0.125$$.

Thus the probabilities of a given allele from your father (again without yet taking into account your own blood type) are

O: $$0.25+0.125/2+0.125/2=0.375$$, A: $$0.3125$$, B: $$0.3125$$.

Given that you inherited either a B or an O, the probability of B is therefore $$\frac{0.3125}{0.3125+0.375}=\frac5{11}$$.

Note that the assumptions in the final point are inconsistent, since if you made the same assumptions about your father's parents you would get a different set of priors for your father's genotype. However, this still leads to the same final answer since the answer only depends on the prevalence of each allele in the population, not the prevalence of each genotype.

• Yeah, it is helpful to read the last point of the clarification – user63710 Nov 27 '19 at 16:54
• Can you double check the case when father is AB? – Manto Nov 28 '19 at 1:54
• @Manto I don't understand. You said that he is equally likely to be any of the four types; that means 25% chance of being AB. What's to check? – Especially Lime Nov 28 '19 at 8:29
• Of course, that's the prior probability; if you want the posteriors for your father's genotype they are $4/11$ for OO, $2/11$ each for AB, BB, BO and $1/11$ for AO. But you don't need to work those out in order to answer the question. – Especially Lime Nov 28 '19 at 8:35

Mom's parents have to be

have to be AB, for them to give her B, but not be B themselves.

Meaning mother is

BB, if she would have got anything else from her parents she would have been AB too.

So options for father are

BB, BO, AB, AO, OO can't be AA, or I would have gotten an A. My options depending on father: BB --> BB BO --> BB/BO AB --> BB AO --> BO OO --> BO

So it seems

That out of 6 outcomes, 3 give me pure blood, which would be 50%.

• Your reasoning is correct, but not the math in the conclusion. The father has 5 different possible blood types each with equal probability ($\frac15$). With each of the father blood type has one or two possible blood type for the son. The probability in this case is $$\frac15 \times 1 + \frac15\times\frac12 + \frac15\times\frac12+\frac15\times0+\frac15\times0=\frac25$$ – Alain Remillard Nov 27 '19 at 17:09
• @AlainRemillard 2/5 = 40% doesn't seem quite right. Check the case when father is AB – Manto Nov 28 '19 at 1:50
• @Manto I haven't notice the fact of I am blood type B Still there are some flaw in the computation since not all the outcome has the same probabillity. – Alain Remillard Nov 28 '19 at 16:40

Because of the codominancy of the mother's parents, the mother is BB pure blood. We only need to calculate with the father this time.

We don't know anything about the fathers parents so the father can be any fenotype with equal probability, from this point of view. But the starting point in the question is that we already know that I am B fenotype, so father cannot be AA genotype (see the last line of the clarification: "the children affecting the parent")

That means the probability of the father genotype AO is actually equal to the probability of his fenotype group B (namely 0.25).

My probability to be "pure blood" = chance to get B allel from father / [chance to get B allel from father + chance to get O allel from father]

=(0.125+0.125+0.0625)/[(0.125+0.125+0.0625)+(0.125+0.25+0.0625)]~0.416

The phrasing in the end of the clarification might sound unclear, though I think no need to state "and the genotype as well" because if there would happen to be an impossible genotype constellation, this becomes obvious.

• 41.6% doesn't seem quite right. Check the case, when father is AB – Manto Nov 28 '19 at 1:49