finding a probability

Question

Suppose a lab is developing a robust new test to look for diabetes in children. If a child actually has diabetes, the probability that the test will be positive is 84%, indicating they have diabetes. If the child does not have diabetes, the probability that the test will be positive is 10%. Suppose 5.6% of all children actually have diabetes. (Round your answer to 3 decimal places, if needed.)

(a) What is the probability that a randomly chosen child has diabetes and tested positive?

(b) What is the probability that a randomly chosen child tested positive?

(c) Given that a test comes up positive, what is the probability the child has diabetes?

(d) What is the probability that a randomly chosen child does not have diabetes and tested negative?

(e) What is the probability that a randomly chosen child tested negative?

(f) Given that a test comes up negative, what is the probability the child does not have diabetes?

Answer 1

a) The probability that a randomly chosen child has diabetes and tested positive can calculated as

P(Diabetes and Positive) = P(Diabetes) * P(Positive | Diabetes) = 0.056 * 0.84 = 0.0469

b) The probability that a randomly chosen child tested positive can be calculated as the sum of the probability of a child with diabetes testing positive and the probability of a child without diabetes testing positive:

P(Positive) = P(Diabetes and Positive) + P(No Diabetes and Positive) = 0.0469 + (1-0.056) * 0.1 = 0.1044

c) Given that a test comes up positive, the probability the child has diabetes can be calculated as

P(Diabetes | Positive) = P(Diabetes and Positive) / P(Positive) = 0.0469 / 0.1044 = 0.448

d) The probability that a randomly chosen child does not have diabetes and tested negative can be calculated as

P(No Diabetes and Negative) = P(No Diabetes) * P(Negative | No Diabetes) = (1-0.056) * (1-0.1) = 0.83944

e) The probability that a randomly chosen child tested negative can be calculated as the sum of the probability of a child with diabetes testing negative and the probability of a child without diabetes testing negative:

P(Negative) = P(Diabetes and Negative) + P(No Diabetes and Negative) = (1-0.84) * 0.056 + 0.83944 = 0.89556

f) Given that a test comes up negative, the probability that the child does not have diabetes can be calculated as

P(No Diabetes | Negative) = P(No Diabetes and Negative) / P(Negative) = 0.83944 / 0.89556 = 0.939