A person can have one of the four blood types A, B, AB, and O. The proportions of the four blood types in a given population are(0.2, 0.3, 0.25, 0.25) respectively. Suppose the likelihood of a certain disease X in a person depends on the person's blood type as follows: A: 0.1, B: 0.1, AB: 0.2, and O: 0.4. a) What is the probability of disease X in a person chosen uniformly at random from the population? b) Knowing that the selected person has disease X, what is the probability that their blood type is A?
Question
A person can have one of the four blood types A, B, AB, and O. The proportions of the four blood types in a given population are(0.2, 0.3, 0.25, 0.25) respectively. Suppose the likelihood of a certain disease X in a person depends on the person's blood type as follows: A: 0.1, B: 0.1, AB: 0.2, and O: 0.4. a) What is the probability of disease X in a person chosen uniformly at random from the population? b) Knowing that the selected person has disease X, what is the probability that their blood type is A?
Solution
a) To find the probability of disease X in a person chosen uniformly at random from the population, we need to multiply the probability of each blood type by the probability of disease X given that blood type, and then sum these products. This is because the total probability of an event is the sum of the probabilities of that event occurring in each of the mutually exclusive scenarios (in this case, the four blood types).
So, the probability of disease X is:
(0.2 * 0.1) + (0.3 * 0.1) + (0.25 * 0.2) + (0.25 * 0.4) = 0.02 + 0.03 + 0.05 + 0.1 = 0.2
b) To find the probability that a person with disease X has blood type A, we use Bayes' theorem, which states that the probability of event A given event B is equal to the probability of event B given event A times the probability of event A, divided by the probability of event B.
In this case, event A is having blood type A, and event B is having disease X. So, the probability of blood type A given disease X is:
(0.1 * 0.2) / 0.2 = 0.02 / 0.2 = 0.1
So, if we know that a person has disease X, there is a 10% chance that their blood type is A.
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