Question

	DT: Donor's Blood Type
		O-	O+	A-	A+	B-	B+	AB-	AB+	P(RT)
RT:Receiver's Blood Type	O-	1	0	0	0	0	0	0	0	7.0%
	O+	1	1	0	0	0	0	0	0	37.0%
	A-	1	0	1	0	0	0	0	0	6.0%
	A+	1	1	1	1	0	0	0	0	36.0%
	B-	1	0	0	0	1	0	0	0	2.0%
	B+	1	1	0	0	1	1	0	0	8.0%
	AB-	1	0	1	0	1	0	1	0	1.0%
	AB+	1	1	1	1	1	1	1	1	3.0%
	P(DT)	7.0%	37.0%	6.0%	36.0%	2.0%	8.0%	1.0%	3.0%	100.0%

Table of Probability of Successful Donation Given DT and RT: P(D|DT,RT)

Each person has a blood type (O-,O+,A-,A+,B-,B+,AB-,AB+). The table above shows the probability of a successful blood transfusion given the donor's blood type (DT) and the receiver's blood type (RT). A "1" in this table means that the blood transfusion from DT to RT would be successful. Conversely, a "0" means that a transfusion from DT to RT would be unsuccessful, meaning that the receiver would die if such a transfusion took place. As you can see, O- donors can give to anyone. Conversely, AB+ receivers can receive blood from anyone.

P(DT) is the probability that a randomly selected donor would have a particular blood type. Likewise, P(RT) is the probability that a randomly selected receiver would have a particular blood type.

(a) What is the probability that a randomly selected donor and receiver would have a successful blood transfusion? (The answer is NOT 27/64)

(b) Suppose someone with an unknown blood type needs a transfusion. While their blood type is unknown, we do know that they underwent a successful blood transfusion from an O+ donor in the past. We have obtained an A- donor. What is the probability that the transfusion would be successful from this donor? (The answer is NOT 25% or 50%)

Answer 1

a.) Answer = 0.5617

P(DT,RT) = P(DT) * P(RT)

This gives us the probability table. Multiplying the probability table with the incidence table i.e. the table of 1 or 0 we will get the probability of successful transfusion which will sum of the probabilities in the table.

b.) As transfusion from an O+ donor was successful, the receiver will be one of the following (O+, A+, B+, AB+).

Now for an A- donor transfusion will be successful if the receiver is A+ or AB+. So probability of successful transfusion will be =(P(A+) + P(AB+)) \ (P(A+) + P(AB+) + P(O+) + P(B+))

= 0.4643