EXAMPLE 8 Arsenic Testing: Using the General Multiplication Rule

Tests, whether medical screening tests for diseases or testing drinking water for health- risk contaminants, are not perfect. Take, for example, testing for arsenic in drinking water. Suppose that a test for arsenic in the water supply correctly reports the presence of arsenic (positive test result) with probability 0.97, and correctly reports the absence of arsenic (negative test result) with probability 0.86. Only about 4% of the drinking water in the United States is thought to contain arsenic.

Using the information above, we want to determine the following two probabilities:

  1. The probability that a randomly chosen water sample actually contains arsenic and yields a positive test result.
  2. The probability that a randomly chosen water sample does not contain arsenic and yields a negative test result.

Next, we translate what we know and what we want to find into mathematical notation. Let be the event that a water sample contains arsenic, and let be the event that the sample does not contain arsenic. Let “+“ represent the event that the test results come back positive for arsenic, and “−” the event that the test results come back negative.

  • What we know: , and
  • What we want to find: and

Using the information from the first bullet and the general multiplication rule, we calculate probabilities 1 and 2 as follows: