More Probability (Dr. Monticino)

1
More Probability

(Dr.
Monticino)
2
Overview

• Conditional probability and independence
• Multiplication and addition rules
• Examples, examples, examples

3
Conditional Probability

• Recall, that conditional probability provides a
  formal way for conditioning probabilities based
  on new information
• In particular,
• P(A B) P(A ? B)/P(B)
• P(A ? B) P(A B) ? P(B)

4
Independence

• Intuitively, two events are independent if
  information that one occurred does not affect the
  probability that the other occurred
• More formally, A and B are independent if
• P(A B) P(A)
• P(B A) P(B)
• P(A ? B) P(A)P(B)

5
Multiplication Rule

• The probability of the intersection of two events
  equals the probability of the first multiplied by
  the probability of the second given that the
  first event has happened
• P(A ? B) P(A B) ? P(B)
• If the two events are independent, then
• P(A ? B) P(A) ? P(B)

6
Addition Rule

• If two events are disjoint, then the probability
  that at least one happens is the sum of the
  probabilities of each event
• P(A ? B) P(A) P(B)
• 1 P(?) P(A ? Ac) P(A) P(Ac)
• So, P(A) 1 - P(Ac)
• Thus, you can only add two probabilities if the
  events are mutually exclusive.

7
Examples

• Draw two cards are drawn from a well-shuffled
  deck of cards
• What is the probability that the first card is
  either a queen or a king?
• Find the probability that the first is a queen
  and the second is a king
• Without replacement
• With replacement
• Find the probability that the second card is a
  king?
• Probability of drawing queen and king

8
Examples

• A coin is tossed three times
• What is the probability of getting no heads?
• What is the probability of getting exactly two
  heads?
• What is the probability of getting at least one
  head?
• Suppose that a coin is tossed 50 times
• Given that the first 49 tosses come up tails,
  what is the probability that the 50th toss is a
  head?
• What is the probability of getting 50 tails in a
  row?

9
Examples

• Paradox of the Chevalier de Mere
• The birthday problem
• (Dr. Monticino)

10
Birthday Probabilities