(Blood Test Problem) Suppose 100 people are waiting for blood test for a kind of disease. The probability that for one person the test result is positive equals 0.1. Two test methods are as follows:
Method 1: Test one by one. Then we have to do 100 tests.
Method 2: Divide 100 people into 25 groups with 4 people in each. Then, mix 4 people’s blood and test. If the result is negative, then everyone in this group is negative. If the result is positive, then we test these 4 people one by one. Now, the number of tests becomes 1+4=5.
Question: Which method is better?
If they are both as accurate, which will involve fewer tests?
To determine which method is better, we need to compare the number of tests required and consider the probabilities involved.
Method 1 requires testing each person individually, resulting in 100 tests.
Method 2 involves dividing the 100 people into 25 groups of 4. If the test for a group comes back negative, then all 4 people in that group are considered negative, resulting in only 1 test for that group. However, if the test for a group is positive, then the 4 individuals in that group need to be tested individually, adding 4 additional tests. So for each group, the maximum number of tests required is 5 (1 group test + 4 individual tests).
To determine the total number of tests needed using Method 2, we need to consider the probability of a positive test result for each group.
The probability of a positive test result for one person is given as 0.1. Therefore, the probability of all 4 people in a group testing negative is (1-0.1)^4 = 0.6561. This means the probability of a group requiring only 1 test is 0.6561.
The probability of at least one person in a group testing positive is 1 - 0.6561 = 0.3439. This means the probability of a group requiring 5 tests is 0.3439.
Using these probabilities, the expected number of tests for Method 2 can be calculated:
Expected number of tests = (number of groups requiring 1 test * 1 test) + (number of groups requiring 5 tests * 5 tests)
Expected number of tests = (25 * 0.6561 * 1) + (25 * 0.3439 * 5)
Expected number of tests = 16.4025 + 42.975
Expected number of tests = 59.3775
Comparing the number of tests required for Method 1 (100) and Method 2 (59.3775), we can conclude that Method 2 is better as it requires fewer tests on average.