SECTION 4 4 MULTIPLICATION RULE BASICS BASICS UNDERSTANDING

SECTION 4. 4 MULTIPLICATION RULE : BASICS

BASICS UNDERSTANDING • P(A and B) • P(A∩B) • Involves the multiplication of the probability of event A and the probability of event B • If necessary, the probability of event B is adjusted because of the outcome of event A.

P(A and B) • P(event A occurs in a first trial and event B occurs in a second trial)

Examples of P(A and B) • Find the probability when a baby is born, it is a girl and when a single die is rolled, the outcome is six. • Find the probability when a day of the week is randomly selected, it is a Saturday and When a second different day of the week is randomly selected, it is a Monday.

TYPES OF MULTIPLICATION RULES • Independent – Will discuss and explain formula in this section • Dependent – Will discuss in this section, but explain formula in section 4. 5

INDEPENDENT Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other. Sampling with replacement is an independent event.

DEPENDENT Two events are dependent if the occurrence affects the probability of the occurrence of the other. • Sampling without replacement is a dependent event.

Independent or Dependent? • Find the probability when a baby is born, it is a girl and when a single die is rolled, the outcome is six. • Find the probability when a day of the week is randomly selected, it is a Saturday and When a second different day of the week is randomly selected, it is a Monday.

INDEPENDENT • Find the probability when a baby is born, it is a girl and when a single die is rolled, the outcome is six.

DEPENDENT • Find the probability when a day of the week is randomly selected, it is a Saturday and When a second different day of the week is randomly selected, it is a Monday.

INDEPENDENT FORMULA P(A and B) = P(A∩B) = P(A) * P(B) A is the first event B is the second event ∩ = intersection = and

INDEPENDENT • Find the probability when a baby is born, it is a girl and when a single die is rolled, the outcome is six.

SOLVE - INDEPENDENT • Find the probability when a baby is born, it is a girl and when a single die is rolled, the outcome is six. • P(Girl and six) = P(Girl) * P(Six) • P(Girl and six) = (1 / 2) * (1 / 6) • P(Girl and six) = 1 / 12

EXAMPLE A Positive Test Results Subject Uses 44 Drugs (True Positive) Subject is not a 90 Drug User (False Positive) Negative Test Results 8 (False Negative) 860 (True Negative)

EXAMPLE A • Given the chart if 2 of the 1000 test subjects are randomly selected, find the probability that they both had false positives if the 2 selections are made with replacement.

EXAMPLE A Given the chart if 2 of the 1000 test subjects are randomly selected, find the probability that they both had false positives if the 2 selections are made with replacement. • with Replacement … Independent • P(A and B) = P(A) * P(B)

EXAMPLE A • Independent … with replacement • P(A and B) = P(A) * P(B) = (90 / 1000) * (90 / 1000) • P(A and B) = P(A) * P(B) = 81 / 10000 • P(A and B) = P(A) * P(B) = 0. 0081

EXAMPLE B 1 • For the following , ignore lead years and that assume that births on the 365 different days of the year are equally likely. • What is the probability that a randomly selected person was born on July 4?

EXAMPLE B 1 • P(born on 7/4) = (July 4)/(year) • P(born on 7/4) = 1 / 365 • P(born on 7/4) = 0. 002739726 • P(born on 7/4) = 0. 0027

EXAMPLE B 2 • For the following , ignore lead years and that assume that births on the 365 different days of the year are equally likely. • What is the probability that two randomly selected people were both born on July 4?

EXAMPLE B 2 P(born on 7/4 and born on 7/4) = P(7/4) ∩ P(7/4) = (1 / 365) * (1 / 365) P(7/4) ∩ P(7/4) = 1 / 133225 P(7/4) ∩ P(7/4) = 0. 000007506098705 P(7/4) ∩ P(7/4) = 7. 506098705 x 10 -6

EXAMPLE B 3 • What is the probability that a randomly selected person were born on the same day?

EXAMPLE B 3 • P (born and on same day) = P(born) ∩ P(same day) • P(born) ∩ P(same day) = (1) * (1 / 365) • P(born) ∩ P(same day) = 1 / 365 • P(born) ∩ P(same day) = 0. 002739726