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($$A^c$$ B) =1

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