Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Multiplication Rule Sections 3-4 & 3-5 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

Finding the Probability of Two or More Selections Multiple selections Multiplication Rule

Example: Tom and Fred are playing a game. One of them is asked to pick a card from a pocket, that contains 5 cards marked a, b, c, d, e. Q: What is the probability of the event A = { Tom picks the “c” card }

FIGURE Tree Diagram of Test Answers

T F FIGURE Tree Diagram of the events T & a T & b T & c T & d T & e F & a F & b F & c F & d F & e T F

T F FIGURE Tree Diagram of Test Answers T & a T & b T & c T & d T & e F & a F & b F & c F & d F & e T F

P(T) = T F FIGURE Tree Diagram of Test Answers T & a T & b T & c T & d F & a F & b F & c F & d F & e T F 1 2 P(T) =

P(T) = P(c) = T F FIGURE Tree Diagram of Test Answers T & a T & b T & c T & d T & e F & a F & b F & c F & d F & e T F 1 2 1 5 P(T) = P(c) =

P(T) = P(c) = P(T and c) = T F FIGURE Tree Diagram of Test Answers b c d e T & a T & b T & c T & d T & e F & a F & b F & c F & d F & e T F 1 2 1 5 1 10 P(T) = P(c) = P(T and c) =

P (Tom and c card)

P (Tom and c card) = P (T and c)

P (Tom and c card) = P (T and c) 1 10 1 1 2 5

P (Tom and c card) = P (T and c) 1 10 1 1 = • 2 5 Multiplication Rule

INDEPENDENT EVENTS = • 1 1 1 10 2 5 P (Tom and c card) = P (T and c) Multiplication Rule INDEPENDENT EVENTS

Definitions Independent Events Dependent Events Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. Dependent Events If A and B are not independent, they are said to be dependent.

Notation for Multiplication Rule P(B | A) represents the probability of B occurring after it is assumed that event A has already occurred (read B | A as “B given A”).

Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.) P(Ace on first card) = P(King/Ace) = P(drawing Ace, then a King) = • = 4 52 4 51 4 52 4 51 16 .00603 ­ 2652 DEPENDENT EVENTS

Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.) P(Ace on first card) = 4 52

Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.) P(Ace on first card) = P(King Ace) = 4 52 4 51

Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.) P(Ace on first card) = P(King Ace) = P(drawing Ace, then a King) = • = 4 52 4 51 4 52 4 51 16 0.00603 » 2652

Example: Find the probability of drawing two cards from a shuffled deck of cards such that the first is an Ace and the second is a King. (The cards are drawn without replacement.) P(Ace on first card) = P(King Ace) = P(drawing Ace, then a King) = • = 4 52 4 51 4 52 4 51 16 0.00603 » 2652 DEPENDENT EVENTS

Formal Multiplication Rule P(A and B) = P(A) • P(B) if A and B are independent

Formal Multiplication Rule P(A and B) = P(A) • P(B) if A and B are independent P(A and B) = P(A) • P(B|A) if A and B are dependent

P(A and B) = P(A) • P(B | A) Figure Applying the Multiplication Rule P(A or B) Multiplication Rule Are A and B independent ? Yes P(A and B) = P(A) • P(B) No P(A and B) = P(A) • P(B | A)

Intuitive Multiplication When finding the probability that event A occurs on one trial and B occurs on 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.

Small Samples from Large Populations If small sample is drawn from large population (if n £ 5% of N), you can treat the events as independent.

Conditional Probability Definition The conditional probability of B given A is the probability of event B occurring, given that A has already occurred.

Conditional Probability Dependent Events P(A and B) = P(A) • P(B|A)

Conditional Probability Dependent Events P(A and B) = P(A) • P(B|A) Formal

Conditional Probability Dependent Events P(A and B) = P(A) • P(B|A) Formal P(B|A) = P(A and B) P(A)

Conditional Probability Dependent Events P(A and B) = P(A) • P(B|A) Formal P(B|A) = Intuitive P(A and B) P(A)

Conditional Probability Dependent Events P(A and B) = P(A) • P(B|A) Formal P(B|A) = Intuitive The conditional probability of B given A can be found by assuming the event A has occurred and, operating under that assumption, calculating the probability that event B will occur. P(A and B) P(A)

Probability of ‘At Least One’

Probability of ‘At Least One’ ‘At least one’ is equivalent to one or more.

Probability of ‘At Least One’ ‘At least one’ is equivalent to one or more. The complement of getting at least one item of a particular type is that you get no items of that type.

Probability of ‘At Least One’ ‘At least one’ is equivalent to one or more. The complement of getting at least one item of a particular type is that you get no items of that type. If P(A) = P(getting at least one), then

Probability of ‘At Least One’ ‘At least one’ is equivalent to one or more. The complement of getting at least one item of a particular type is that you get no items of that type. If P(A) = P(getting at least one), then P(A) = 1 – P(A)

Probability of ‘At Least One’ ‘At least one’ is equivalent to one or more. The complement of getting at least one item of a particular type is that you get no items of that type. If P(A) = P(getting at least one), then P(A) = 1 – P(A) where P(A) is P(getting none)