Abstract
In order to account for multiple variables that may impact the probability of an event Bayes Theorem may be applied accordingly. Subsequently, Bayes Theorem of conditional probability can be applied to predict the possibility of deriving one certain outcome during an event. One scientific article chronicles the “Causes of Outbreaks Associated with Drinking Water in the United States from 1971 to 2006”.

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Introduction
Supposing that three (3) governing agencies charged with the protection of health in the United States were investigating drinking water related diseases and included; the Centers for Disease Control (CDC), the Environmental Protection Agency (EPA) and the Council of State and Territorial Epidemiologists (CSTE). Unaware of which particular group has general oversight it can be assumed that there is likely to be a 33% chance or probability that the CDC is taking charge of this health related study. However, if one is told that the agency that is leading the study is likely to be governed only by federal authority the CSTE can be ruled out of its leadership role. Bayes theorem can therefore be utilized to calculate the probability that the CDC is leading this study.

Applying the Principles of Conditional Probability
W will represent the event that the study is being led by the CDC, and L will represent the event that the study pertains to drinking water. Using historical data, it can be further assumed that the CDC has led large undertakings of this nature for 1/2 of all executed studies. The probability of W occurring is therefore P (W) = 0.5. If it is also known that 85% of all drinking water related studies are led by the CDC, this can be annotated by P (L/W) = 0.85. This means that, the probability of L given W is 0.85. Since we already know the study is about drinking water, event L is 75%. Furthermore, if we know that 20% of health related studies are led equally by both federal groups, then this is considered a complementary event (assuming that the CSTE is ruled out, then the study can only be led by CDC, or EPA). This will be represented by P(L/M) = 0.20.

Objective:

Determine the probability that the study was conducted by the CDC alone.

P(W/L) = P(L/W) x P(W) / P(L)

P (W/L) = (0.85 X 0.50) / (0.85 X 0.50) + (0.20 X 0.50) = .809 or 81%.

Conclusion
There is an 81% Chance that the Study was led by the CDC.