Independence

A property where two events do not influence each other, meaning the occurrence of one tells us nothing about the other. 🔗

Independent Events

Suppose A and B are two events.

Two events are independent if the occurrence of event B provides no information about the occurrence of event A:

P(A|B) = P(A)

Another definition of independence:

P(A \cap B) = P(A) P(B)

For multiple events:

P(A_{1} \cap A_{2} \cap ... A_{n}) = P(A_{1}) P(A_{2}) ... P(A_{n})

Suppose A, B, and C are events.

If A and B are independent, conditioning on C may remove that independence.

When we condition on C, events A and B may no longer be independent.

Exercise

The king comes from a family of two children. What is the probability that his sibling is female F and not male M?

Let all outcomes be equally likely.

Solution

Sample \ Space = \{(FF), (FM), (MF), (MM)\} = \{(\not F \not F), (FM), (MF), (MM)\}

P(F|M) = \frac{P(F \cap M)}{M} = \frac{2}{3} \approx 0.6\bar{6}