# Copyright © Ed2Net Learning Inc.1. 2 Warm Up Use the Counting principle to find the total number of outcomes in each situation 1. Choosing a car from.

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2 Warm Up Use the Counting principle to find the total number of outcomes in each situation 1. Choosing a car from five different exterior colors 40 2. Making a sandwich with raisin bread, whole wheat bread, white bread, or bagel and choosing peanut butter, cream cheese, or jelly. 12 3. Choosing a 4-digit PIN if the numbers can be repeated. 10,000 4. Choosing the first two characters of a license plate if it begins with a letter of the alphabet and is followed by a digit. 260 5. Flipping a penny, nickel and a dime. 8

Copyright © Ed2Net Learning Inc.3 Event : An Event is an experiment. Outcome: Possible outcomes of an event are the results which may occur from any event. Lets review what we have learned in the last lesson Counting principle: Counting principles describe the total number of possibilities or choices for certain selections.

Copyright © Ed2Net Learning Inc.4 Permutation: A permutation is an arrangement. Permutations are about Ordering. The formula is nPr = (n!) /(n - r)! Combination: Combination means selection of things. Order of things has no importance. The formula is nCr = (n!) /(r! (n -r)!)

Copyright © Ed2Net Learning Inc.5 What are independent events?: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. Examples: Landing on tails after tossing a coin AND rolling a 6 on a single 6-sided dice. Choosing a ball from a box AND landing on tails after tossing a coin. Lets get Started

Copyright © Ed2Net Learning Inc.6 Probability of Independent Events If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events P(A and B) = P(A) x P(B).

Copyright © Ed2Net Learning Inc.7 Example Example 1: A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the tail side of the coin and rolling a 5 on the die. These events are independent. P(Tail) = 1/2 P(5 on dice) = 1/6 P = P(Tail) X P(5 on dice) = ½ X 1/6 = 1/12 The probability is 1/12. P(A and B) = P(A) x P(B).

Copyright © Ed2Net Learning Inc.8 Example Example 2: A fair die is tossed twice. Find the probability of getting a 2 or 3 on the first toss and a 5 in the second toss. These events are independent. P(2 or 3) = 2/6 = 1/3 P(5) = 1/6 P = P(2 or 3) X P(5) = 1/3 X 1/6 = 1/18 The probability is 1/18. P(A and B) = P(A) x P(B).

Copyright © Ed2Net Learning Inc.9 What are dependent events?: Two events, A and B, are dependent if the fact that A occurs does not affects the probability of B occurring. Examples: Choosing a sock from a box and then choosing another from that box without replacing the first one. Choosing a ball from a box AND choosing another ball from that box.

Copyright © Ed2Net Learning Inc.10 Probability of Dependent Events If A and B are dependent events, the probability of both events occurring is the product of the probability of the first event and the probability of the second event once the first event has occurred. P(A and B) = P(A) x P(B, once A has occurred )

Copyright © Ed2Net Learning Inc.11 Example Example 3: A box contains a quarter, a dime and a penny. Find the probability of choosing a quarter first and then choosing a penny without replacing quarter. These events are dependent. P (quarter) = 1/3 P (penny) = 1/2 P = P (quarter) X P (penny) = 1/2 X 1/3 = 1/6 The probability is 1/6. P(A and B) = P(A) x P(B|A)

Copyright © Ed2Net Learning Inc.12 Example Example 4: A box contains two red balls and four blue balls. Find the probability of choosing a red ball first and then choosing a blue ball without replacing red ball. These events are dependent. P (red ball) = 2/6 = 1/3 P (blue ball) = 4/5 P = P (red ball) X P (blue ball) = 1/3 X 4/5 = 4/15 The probability is 4/15. P(A and B) = P(A) x P(B|A)

Copyright © Ed2Net Learning Inc.13 Your Turn Tell whether the events are independent or dependent: 1.Rolling a dice and choosing a card from deck of cards Answer= Independent 2.Rolling a dice and Tossing a coin Answer= Independent 3.Choosing a card and choosing another card from the same deck of cards Answer= Dependent 4.Roll a cube and spin a spinner Answer= Independent

Copyright © Ed2Net Learning Inc.14 Your Turn Find the probability: 5.Choosing red marbles from a box one after another Answer= Dependent 6.Flip a coin and toss a 1-6 number cube. P (head and 4) Answer= 1/12 7.Choosing a heart and spade from a deck of cards Answer= 13/204 8.There are 2 orange, 5 red in a box. You pick 2 marbles from the hat. Marbles are not returned after they have been drawn. P (the first marble is orange and the second marble is red) Answer= 5/21

Copyright © Ed2Net Learning Inc.15 9. 9.A deck of cards has 2 navy and 3 red cards. You pick 2 cards from the deck. Cards are not returned to the deck after they are picked. P (the first card is navy and the second card is red) Answer= 3/10 10. 10.Find the probability of rolling a cube which has the numbers 6, 8, 10, 13, 16, and 17 on it. You then spin a spinner which has 3 sections. The letters on the spinner are A, C, and B. P(10 and B) Answer= 1/18 Your Turn

Copyright © Ed2Net Learning Inc.18 Q1. A bag contains 5 white marbles, 3 black marbles and 2 green marbles. In each draw, a marble is drawn from the bag and not replaced. Find the probability of obtaining white, black and green in that order. These events are dependent. There are total 10 marbles in the bag. P (white) = 5/10 = 1/2 P (black) = 3/9 = 1/3 P (green) = 2/8 = 1/4 P = P (white) X P (black) X P (green) = 1/2 x 1/3 x 1/4 = 1/24 The probability is 1/24.

Copyright © Ed2Net Learning Inc.19 Q2. You choose a card from a deck of cards, one at a time, without replacing them. What is the probability that you get an ace then a king? These events are dependent. P (Ace) = 4/52 = 1/13 P (King) = 4/51 P = P (Ace) X P (King) = 1/13 x 4/51 = 4/663 The probability is 4/663.

Copyright © Ed2Net Learning Inc.20 Q3. You choose a card from a deck of cards, one at a time, without replacing them. What is the probability that you get three queens in a row? These events are dependent. P (1 st queen) = 4/52 = 1/13 P (II nd queen) = 3/51 = 1/17 P (III rd queen) = 2/50 = 1/25 P = P (1 st queen) X P (II nd queen) X P (III rd queen) = 1/13 x 1/17 x 1/25 = 1/5525 The probability is 1/5525.

Copyright © Ed2Net Learning Inc.21 Let Us Review If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events P(A and B) = P(A) x P(B). If A and B are dependent events, the probability of both events occurring is the product of the probability of the first event and the probability of the second event once the first event has occurred. P(A and B) = P(A) x P(B, once A has occurred )

Copyright © Ed2Net Learning Inc.22 You did great Today!! Be sure to keep practicing

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