Example 10

EXAMPLE 10 Arsenic Testing: How Worried Should You be about a Positive Test Result?

We begin with a tree diagram, shown in Figure 8.15, to identify the sample space of all possible outcomes, and use the general multiplication rule (Rule 7) to assign probabilities to each outcome in the sample space. Two of these probabilities were already computed in Example 8. In addition, the tree diagram contains the conditional probability models from Example 9 and Self Check 5.

Figure 8.15: Figure 8.15 Tree diagram of sample space and calculation of probabilities.

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We want , which by the definition of conditional probability we calculate as follows:

From the tree diagram, we know . Next, we need . There are two ways to get a positive result: or . Since these are disjoint events, we use the addition rule for disjoint events (Rule 4) to find :

Next, we substitute and into the conditional probability formula:

In this situation, only around 22% of the water samples that test positive for arsenic actually contain arsenic. So, until further testing is done, you probably shouldn’t be in total panic mode. In Self Check 6, you will see how this probability changes when the likelihood of having arsenic in the water supply is increased.