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Elementary Statistics: Multiplication Rule Basics

Learn how to find the probability of two or more selections using the multiplication rule in statistics. Understand independent and dependent events, formal multiplication rule, and examples involving coin flips, dice rolls, and ball selections.

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Elementary Statistics: Multiplication Rule Basics

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  1. STATISTICS ELEMENTARY Section 3-4 Multiplication Rule: Basics MARIO F. TRIOLA EIGHTH EDITION

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

  3. Notation P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)

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

  5. Definitions • Independent 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.

  6. Formal Multiplication Rule • P(A and B) = P(A) • P(B A) • If A and B are independent events, P(B A) is really the same as P(B)

  7. Figure 3-10 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)

  8. Intuitive Multiplication 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 • A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die. • Are the independent events? Yes • Solution: • P(Head and 4) = P(Head) * P(4) = 1/2 * 1/6 = 1/12

  10. Examples • A bag contains 3 red balls, 2 blue balls and 5 white balls. Balls are selected with replacement in between. • Are the independent events? Yes because of the replacement. • P( Red and Blue)? = P(Red) * P(Blue) = 3/10 * 2/10 = 6/100 = 3/50 =.06 • P( Red and White and Blue)? = P(Red)*P(White)*P(Blue) = 3/10 *5/10* 2/10 = 30/1000 = 3/100 = .03

  11. Examples • A bag contains 3 red balls, 2 blue balls and 5 white balls. Balls are selected without replacement in between. • Are the independent events? No because there are less balls each time. • P( Red and White)= = P(Red) * P(White|Red) = 3/10 * 5/9 = 15/90 = 1/6 =.167 • P( Red and White and Blue) = = P(Red)*P(White|Red)*P(Blue|Red & White) = 3/10 *5/9* 2/8 = 30/720 = 1/24 = .0417 • P( Red and Red)= = P(Red and Red) = P(Red)*P(Red|Red) = 3/10 *2/9 = 6/90 = 1/15 = .0667

  12. STANDARD DECK OF CARDS • Standard Deck Consists of 52 Cards • 4 Suits: • Hearts(♥), Clubs(♣), Diamonds(♦), Spades(♠) • 2 Colors: 26 of each color • Red: Hearts and Diamonds, Black: Clubs and Spades • 13 Cards in each Suit: • Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen and King • 3 Face Cards in each Suit • Jack, Queen and King • 4 of each card • Ex. There are 4 Aces, 4 Kings, etc.

  13. Examples • Two cards are selected from a deck without replacement in between. • Are the independent events? No because there are less cards each time. • P( Heart and Club)= = P(Heart) * P(Club|Heart) = 13/52 * 13/51 = 169/2652 =.0637 • P( Ace and Ace)= = P(Ace)*P(Ace|Ace) = 4/52 *3/51 = 12/2652 = .00452

  14. Small Samples from Large 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).

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