Is it independent?

Question

Let there be two tests for high blood sugar performed by two different hospitals, A and B. Blood samples are taken from people are first sent to Hospital A and then Hospital B. Hospital A does not reveal the outcome of its tests to Hospital B.

Suppose that the tests are based on similar biological mechanisms. Let A+ = Event that A declares blood sample has high blood sugar and B+ = Event that B declares blood sample has high blood sugar

Are A+ and B+ independent or dependent?

I think that it's independent because Hospital A does not reveal the outcome of its tests to Hospital B.

However, does the fact that blood samples are taken from people are first sent to Hospital A and then Hospital B affect its independence?

Answer 1

The events are absolutely not independent, since if you know the first test came back positive, that significantly increases the chances that the second one will come back positive. Remember that "dependence" does not imply any kind of causal relationship, just a statistical one.

Strictly speaking, your question is impossible to answer because it's rather vaguely stated - you would need to specify a probability distribution etc etc in order to actually be able to answer the question mathematically. However, for a reasonable interpretation of your scenario, the answer is that the two events are dependent.

For example, let's say a blood test works like this:

• If the person has high blood sugar, the test comes back positive with probability $0.9$.
• If the person has normal blood sugar, the test never comes back positive.

Let's also say that 5% of the population has high blood sugar. Well then with no information about anything, the probability that even $B$ will occur (the second test comes back positive) is $0.05\cdot 0.9=0.045$, so a 4.5% chance. However, if you know the first test came back positive, that means the person must have high blood pressure, which means the probability that the second test comes back positive is suddenly 90% rather than just 4.5%.

Answer 2

Two events are said to be independent when the outcome of one event does not affect the outcome of the other event

A+ and B+ are independent of each other

Reason : It is given that hospital A does not reveal any test result to hospital B so the decision made by hospital B is not dependent on the decision made by hospital A . The fact that the test results are given to A first is immaterial as decision made by B is not influenced by the decision made by A.