Multiplication Rule: Basics

1
Section 4-4

• Multiplication Rule Basics

2
NOTATION
P(A and B) P(event A occurs in a first trial
and event B occurs in a second trial)
3
EXAMPLES

• Suppose that you first toss a coin and then roll
  a die. What is the probability of obtaining a
  Head and then a 2?
• A bag contains 2 red and 6 blue marbles. Two
  marbles are randomly selected from the bag, one
  after the other, without replacement. What is the
  probability of obtaining a red marble first and
  then a blue marble?

4
CONDITIONAL PROBABILITY

• If event B takes place after it is assumed that
  event A has taken place, we notate this by BA.
  This is read B, given A.
• P(BA) represents the probability of event B
  occurring after it is assumed that event A has
  already occurred.

5
INDEPENDENT AND DEPENDENT EVENTS

• Two events A and B are independent if the
  occurrence of one event does not affect the
  probability of the occurrence of the other.
• Several events are independent if the occurrence
  of any does not affect the occurrence of the
  others.
• If A and B are not independent, they are said to
  be dependent.

6
FORMAL MULTIPLICATION RULE
P(A and B) P(A) P(BA)
NOTE If events A and B are independent, then
P(BA) P(B) and the multiplication rule
simplifies to P(A and B) P(A) P(B)
7
APPLYING THE MULTIPLICATION RULE
8
INTUITIVE MULTIPLICATION RULE
When finding the probability that event A occurs
in one trial and B occurs in the next trial,
multiply the probability of event A by the
probability of event B, but be sure that the
probability of event B takes into account the
previous occurrence of event A.
9
EXAMPLES

• What is the probability of drawing an ace from
  a standard deck of cards and then rolling a 7
  on a pair of dice?
• In the 105th Congress, the Senate consisted of 9
  women and 91 men, If a lobbyist for the tobacco
  industry randomly selected two different
  Senators, what is the probability that they were
  both men?
• Repeat Example 2 except that three Senators are
  randomly selected.

10
EXAMPLE
In a survey of 10,000 African-Americans, it was
determined that 27 had sickle cell anemia.

• Suppose we randomly select one of the 10,000
  African-Americans surveyed. What is the
  probability that he or she will have sickle cell
  anemia?
• If two individuals from the group are randomly
  selected, what is the probability that both have
  sickle cell anemia?
• Compute the probability of randomly selecting two
  individuals from the group who have sickle cell
  anemia, assuming independence.

11
SMALL SAMPLES FROMLARGE POPULATIONS
If a sample size is no more than 5 of the size
of the population, treat the selections as being
independent (even if the selections are made
without replacement, so they are technically
dependent).