The blood type distribution in the United States is type A 4

The blood type distribution in the United States is type A, 41%; type B, 9%; type AB. 4%; and type 0, 46%. It is estimated that during World War II, several errors were made in typing blood. In fact, 4% of inductees with type O blood were listed as having type A; 88% of those with type A were correctly listed; 4% with type B blood were listed as type A; and 10% with type AB blood were listed as type A A soldier was wounded and brought to surgery He was listed as having type A blood. What is the probability that this is his true blood type? (YOU MUST USE BAYE?S RULE TO RECEIVE CREDIT FOR THIS ASSIGNMENT)

Solution

First we must to calculate the initial probability in fractions, so we have:

Type A=0,41% ====>41/100
Type B=0,09% ====> 9/100
Type AB=0,04% ====>4/100
Type O=0,46% ====>46/100
Total =============>100/100

Then, we have that all the problem is concetrated in the type A and we have that the probability of this type is 41/100.

To calculate the percentage of blood type O that was listed as A we have to multiplicate the 4%=0.04=4/100 to the probability of A
(4/100)x(41/100)=0.0164

To calculate the percentage of blood type B that was listed as A we have to multiplicate the 4%=0.04=4/100 to the probability of A
(4/100)x(41/100)=0.0164

To calculate the percentage of blood type AB that was listed as A we have to multiplicate the 10%=0.10=10/100 to the probability of A
(10/100)x(41/100)=0.04

then we have that 88% of people listed as type A is right

(41/100)x(88/100)=0.3608

Then we must substract the previous probabilities to 0.3608

0.0164+0.0164+0.04=0.0728

0.3608-0.0728=0.288

We have 28% that this is the true blood type