Two events are independent if they do not influence each other.

Events A and B are independent
if and only if P (A ∩ B) = P (A) P (B)

Conditional Probability

For any two events A and B with P(B) ˃ 0,
the conditional probability of A given B has occurred is defined by
P (A | B) = \(\frac{P (A ∩ B)}{P (A)}\)

P (A ∩ B) = P (A | B)P (A)

Then A and B are independent
if and only if either of the statements is true
P (A) = P (A | B)

Terminology

• A false positive results when a test indicates a positive status when the true status is negative (T | \(D^c\))
• A false negative results when a test indicates a negative status when the true status is positive (\(T^c\) | D)
• The Sensitivity(true positive rate )of a test is a probability of a positive test result given the presence of the disease P(T | D).
• The Specificity (true negative rate) of a test is a probability of a negative test result given the absence of the disease P(\(T^c\) | \(D^c\))

Sensitivity is complementary to the false negative rate.
P(T | D) + (\(T^c\) | D) = 1
• Specificity is complementary to the false positive rate.
ccc P(\(T^c\) | \(D^c\))+(T | \(D^c\))= 1