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