Conditional probability for medical testing in a forest

Here, we denote by $\sim A$ the event NOT $A$ .

Example 1: Squirrel Flu Testing

In a forest, a group of squirrels is concerned about a new illness called “Squirrel Flu.” It’s more dangerous than the ordinary flu. From the data collected, they know that

Probability of a squirrel having Squirrel Flu $P(SF)= 0.1$
Probability of testing positive if a squirrel has Squirrel Flu $P(T|SF)=0.9$
Probability of testing positive if a squirrel does not have Squirrel Flu $P(T|\sim SF)= 0.2$

Now, A squirrel named Nutty takes the test and it comes back positive. What is the probability that Nutty actually has Squirrel Flu?

Answer:

Using Bayes’ Theorem:

$P(SF|T) = \frac{P(T|SF) \cdot P(SF)}{P(T|SF) \cdot P(SF) + P(T|\sim SF) \cdot P(\sim SF)}$

Now:

$P(T|SF) = 0.9$
$P(SF) = 0.1$
$P(T|\sim SF) = 0.2$
$P(\sim SF) = 1 - P(SF) = 0.9$

Hence,

$P(SF|T) = \frac{0.9 \cdot 0.1}{0.9 \cdot 0.1 + 0.2 \cdot 0.9} = \frac{0.09}{0.09 + 0.18} = \frac{0.09}{0.27} = \frac{1}{3} \approx 0.333$

So, Nutty has about a 33.3% chance of actually having Squirrel Flu upon testing positive.

Example 2: Deer Allergy Testing

In the forest, some deer have allergies to a particular type of flower.

Probability of a deer having allergies $P(A)= 0.15$
Probability of the test detecting allergies if the deer has them $P(T|A)= 0.85$
Probability of the test being positive if the deer does not have allergies $P(T|\sim A)=0.05$