For a class I am intending to use Bayes Theorem to solve a problem. The matter is that there are much easier ways to solve the problem and I don't know how to find the quantities to apply Bayes Theorem.
Here is the problem:
The blood type distribution in the United States is type A, 41%; type B, 9%; type AB, 4%; and type O, 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?
To solve the problem, I multiplied the falsely listed blood type groups and added them, then divided by total probability:
(0.41 x 0.88) (A listed as A)+ (0.9 x 0.04) (B listed as A) + (0.04 x 0.1) (AB listed as A) + (0.46 x 0.04) (O listed as A) = 0.3868 A listed as A is 0.3608, so 0.3608 / 0.3868 = 0.933. 0.933 is the probability.
But what of Bayes' Theorem for the solution? I came up with this much more convoluted response:
Let D denote that you are type A, let C denote that you are correctly listed: P(D|C) is desired. P(D|C) = P(C|D) * P(D)/ (P(C|D) * P(D) + P(C|D') * P(D')) P(C|D) = P(D&C)/P(D) = 0.41 * 0.88 / 0.41 = 0.88 P(D) = 0.41 P(C|D') = 0.59 * 0.88 / 0.59 = 0.88 P(D') = 0.59 So, P(D|C) = 0.88 * 0.41 / ((0.88 * 0.41) + (0.88 * 0.59)) = 0.41 0.41 is the probability.
Obviously very different than my other answer.... and I am confident is my first answer vs. the Bayes' Theorem output.
What am I doing that make my Bayes' Theorem output wrong? Why can't I correctly apply the theorem?