8/25/2017

Example 2.2 Revisit

In a certain population, 10% of the population are rich, 5% are famous, and 3% are both.

  • What is the probability a randomly chosen person is not rich?
  • What is the probability a randomly chosen person is rich but not famous?
  • What is the probability a randomly chosen person is either rich or famous?
  • What is the probability a randomly chosen person is either rich or famous but not both?
  • What is the probability a randomly chosen person has neither wealth nor fame?

Definitions

  • Mutually Exclusive(disjoint): \[A\cap B = \emptyset\]
  • Exhaustive: \[A \cup B = S\]
  • Partition:
    • \(A\cap B = \emptyset\)
    • \(A \cup B = S\)

Conditional Probability

Definition of Conditional Probability

  • The conditional probability of event \(A\), given event \(B\) is written as \(P(A|B)\)

  • In general, if event \(B\) has nonzero probability (\(P(B) > 0\)), then the conditional probability is defined as \[P(A|B) = \frac{P(A\cap B)}{P(B)}\]

Example 3.1

Roll a pair of dice. Given that the two dice have different values, find the probability that the sum of the dice is an even number.

Example 3.2

We will toss a fair coin 3 times. Let event:

  • A = exactly 2 heads;
  • B = \(1_{st}\) toss is heads;
  • C = \(2_{nd}\) toss is heads

Find \(P(A), P(B), P(A\cap B), P(B|A), P(A|B)\) and \(P(B|C)\)

Another Useful form

\[P(A|B) = \frac{P(A\cap B)}{P(B)} = \frac{\frac{N(A\cap B)}{N(S)}}{\frac{N(B)}{N(S)}} = \frac{N(A\cap B)}{N(B)}\]

Distributive Laws

For events \(A_1\), \(A_2\) and \(B\)

  1. \((A_1 \cup A_2) \cap B = (A_1 \cap B) \cup (A_2 \cap B)\)
    • Distributive laws of multiplication \[(A_1 + A_2) \times B = A_1 \times B + A_2 \times B\]
  2. \((A_1 \cap A_2) \cup B = (A_1 \cup B) \cap (A_2 \cup B)\)

Distributive Laws

For any events \(A_1, A_2, \cdots\)

  1. \((\bigcup_i A_i) \cap B = \bigcup_i(A_i \cap B)\)

2 .\((\bigcap_i A_i) \cup B = \bigcap_i(A_i \cup B)\)

Recap

  • Definitions: Random Experiment, Potential Outcome, Element \(\omega\), Event, empty set \(\emptyset\), sample space \(S\), Union \(\cup\), Intersection \(\cap\)

  • Probability Rules
    1. For each event A, \(0\le P(A) \le 1\)
    2. For the sample space S, \(P(S) = 1\)
    3. If \(A_1, A_2, \cdots\) is a collection of disjoint events, then \(P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^\infty P(A_i)\)
  • DeMorgan's law
    1. \((A\cup B)^c = A^c \cap B^c\)
    2. \((A\cap B)^c = A^c \cup B^c\)
  • Conditional Probability: \(P(A | B) = \frac{P(A\cap B)}{P(B)}\)