Independence of Events

Independence of Events

Concept of Independence

• Independence is not solely determined by whether events overlap.

• It depends on the extent of the overlap and the individual probabilities of the events.

Examples Illustrating Dependence

• Upper Left Corner (Complete Dependence): If pink is true, blue is guaranteed. P(Blue|Pink) = 1, which is not equal to P(Blue).

• Lower Right Corner (No Overlap): Events are mutually exclusive. P(Blue|Pink) = 0, but P(Blue)
  eq 0 and P(Pink)
  eq 0.

Independence as a Rarity

• Independence is a specific condition, not commonly found.

Numerical Example

• Consider 100 boxes where blue occupies 30 boxes (P(Blue) = 0.3), and pink occupies 20 boxes (P(Pink) = 0.2).

Condition for Independence

• For independence, P(Pink bigcap Blue) = P(Pink) * P(Blue).

• In the example, P(Pink bigcap Blue) = 0.2 * 0.3 = 0.06, meaning the overlap must be exactly 6 boxes out of 100.