Introduction to Probability Models

Lecture 4

Qi Wang, Department of Statistics

Aug 28, 2017

Reminders

  1. The first quiz will be on this Wednesday
  2. The first homework is due on September 8
  3. Class participation will account for 2.3% of the final score

Questions

  • How to get the slides?
  • Why \(P(A) = 1 - P(A^c)\)

Multiplication Rule

Multiplication Rule

  • \[P(A|B) = \frac{P(A\cap B)}{P(B)}\]
  • \[P(A\cap B) =P(B) \times P(A|B)\]
  • The probability that Events A and B both occur is equal to the probability that Event B occurs times the probability that Event A occurs, given that B has occurred

Example 1

A box contains 6 white balls and 4 red balls. We randomly (and without replacement) draw two balls from the box. What is the probability that the second ball selected is red?

a standard deck

Example 2

Three cards are dealt successively at random and without replacement from a standard deck of 52 playing cards. What is the probability of receiving, in order, a king, a queen, and a jack?

General addition rule for 3 sets

\[\begin{aligned} P(A\cup B \cup C) & = P(A) + P(B) + P(C) \\ & - P(A\cap B) - P(A\cap C) - P(B\cap C) \\ & + P(A\cap B\cap C) \end{aligned} \]
\[P(A\cup B\cup C)\]
\[P(A)\]
\[P(A) + P(B)\]
\[P(A) + P(B) + P(C)\]
\[P(A) + P(B) + P(C) - P(A\cap B)\]
\[P(A) + P(B) + P(C) - P(A\cap B) - P(A\cap C)\]
\[P(A) + P(B) + P(C) - P(A\cap B) - P(A\cap C) - P(B\cap C)\]
\[\begin{aligned} & = P(A) + P(B) + P(C) \\ & - P(A\cap B) - P(A\cap C) - P(B\cap C) \\ & + P(A\cap B\cap C) \end{aligned} \]