I NEED YOUR HELP

5. Your dad only makes one meal for dinner The probability that he makes pizza tonight is 30%. pasta? The probability that he makes pasta tonight is 60%. What is the probability that he makes pizza or A.30% B. 40% C. 60% D. 90% 6. Out of 5200 households surveyed, 2107 had a dog, 807 had a cat, and 303 had both a dog and a cat. What is the probability that a randomly selected household has a dog or a cat? A. 2611 B. 2107 807 5200 D 302 5200 5200 5200 7. A bag contains 6 black marbles, 9 blue marbles, 4 yellow marbles, and 2 green marbles. A marble is randomly selected, replaced, and a second marble is randomly selected. Find the probability of selecting a black marble, then a yellow marble. A. B. + C. 24 D. 147 8. A box of chocolates contains 10 milk chocolates, 8 dark chocolates, and 6 white chocolates. Hanissa randomly chooses a chocolate, eats it, and then randomly chooses another chocolate. What is the probability that Hanissa chose a milk chocolate, and then, a white chocolate? A. 46 B. 10 24 C. 23 D. 552 9. A rental agency has 12 white cars, 8 gray cars, 6 red cars, and 3 green cars for rent. Mr. Escobar rents a car, returns it because the radio is broken, and gets another car. What is the probability that Mr. Escobar is given a green car and then a gray car? 24 A. C. D . 203 10. A bag of jelly beans contains 10 red, 6 green, 7 yellow, and 5 orange jelly beans. What is the probability of randomly choosing a red jelly bean, replacing it, randomly choosing another red jelly bean, replacing it, and then randomly choosing an orange jelly bean? 500 D 250 A. 125 B. 1060 C. 21952 10976 5480 5488 D. Suggested Enrichment/Reinforcement Activitylies Answer the following questions: 1. Differentiate dependent event from an independent event. 2. Describe a situation in your life that involves dependent and independent events. Explain why the events are dependent or independent.Probability of the Union of Two Events Events - a set of possible outcomes resulting from a particular experiment. For example, a possible event when a single six-sided die is rolled is (5, 6), that is, the roll could be a 5 or a 6. In general, an event is any subset of a sample space (including the possibility of an empty set). Union of Events -a set that contains all of the elements that are in at least one of the two events. The union is written as AUB. Intersection of Events - a set that contains all of the elements that are in both events. The intersection of events A and B is written as AnB. A graphical representation of events that is very useful is the Venn diagram. The sample space S is represented as consisting of all the points in a large rectangle, and events are represented as consisting of all the points in circles within the rectangle. Events of interest are indicated by shading appropriate regions of the diagram. Consider a sample space with events A and B. Recall that the union of events A and B is an event that includes all the outcomes in either event A, event B, or both. The symbol u represents union. Below, AUB is shaded. How do you find the number of outcomes in a union of events? If you find the sum of the number of outcomes in event A and the number of outcomes in event B. you might have counted some of the outcomes twice. Therefore, in order to correctly count the number of outcomes in the union of two events, you must count the number of outcomes in each event separately and subtract the number of outcomes shared by both events (so these are not counted twice) P(ALB) = P(A)+P(B)-P(AnB). This is called the Addition Rule for Probability. PROBABILITY OF THE UNION OF TWO EVENTS The probability of the union of two events A and B (written AuB) equals the sum of the probability of A and the probability of B minus the probability of written as AnB ). A and B occurring together (which is called the intersection of A and B and is P(AUB)=P(A) + P(B ) -P(AnB)Example 1 Suppose that in your class of 30 students, 8 students are in band, 15 students play a sport, and 5 students are both in band and play a sport. Let A be the event that a student is in band and let B be the event that a student plays a sport. Create a Venn diagram that models this situation. In order to fill in the Venn diagram, remember that the total of the numbers in circle A must be 8 a 5. and the total of the numbers in circle B must be 15. The intersection of the two circles must contain S 12 P(A B) is the probability that a student is in band or plays a sport or both. With the help of the Venn diagram, this is not too difficult to calculate: P(AUB ) = 3+10+5 -18 30 30 You could also compute this probability using the Addition Rule or the formula: P(AUB)=P(A)+P(B)-P(AnB) 15 5 = 18 30 30 30 30 Example 2. A card is drawn from a standard deck. Find the probability of drawing a heart or a 7. A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart is or -. Since there are four 7's in a standard deck of 52 cards, the probability of drawing a 7 is or -. The only card in the deck that is both a heart and a 7 is the 7 of hearts, so the probability of drawing both a heart and a 7 is - Substitute P(H)= P(7)= ,and P(Hn7)= into the formula. P(AUB)= P(A)+P(B)-P(AnB) 13 - 32 The probability of drawing a heart or a 7 is - Example 2. Suppose the spinner in Figure 2 is spun. We want to find the probability of spinning orange or spinning a b. b Figure 2 There are a total of 6 sections, and 3 of them are white. So, the probability of spinning white is = = -. There are a total of 6 sections, and 2 of them have a b. So, the probability of spinning a b is - -. If we add these two probabilities, we would be counting the sector that is both white and a b twice. To find the probability of spinning a white or a b, we need to subtract the probability thatthe sector is both white and has a b. So, the probability of spinning and white or a b is -, substitute P(white)+P(b)-P(both white and has b), thus we have The probability of spinning white or a b is . Probability of Dependent and Independent Events Dependent events influence the probability of other events - or their probability of occurring is affected by other events. Independent events do not affect one another and do not increase or decrease the probability of another event happening An event is deemed independent when it isn't connected to another event, or its probability of happening, or conversely, of not happening. This is true of events in terms of probability, as well as in real life, which, as mentioned above, is true of dependent events as well. For example, the color of your hair has absolutely no effect on where you work. The two events of "having black hair" and "working in Allentown" are completely independent of one another. Independent events don't influence one another or have any effect on how probable another event is. Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other If two events, A and B, are independent, then the probability of both events occurring is the product of the probability of A and the probability of B. In symbols, P(A and B)=P(A) . P(B) When the outcome of one event affects the outcome of another event, they are dependent events. If two events A and B are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs. In symbols, P(A and B) =P(A) .P(B following A) Situation 1: Consider a box that contains 14 red balls, 12 blue balls, and 9 yellow balls. A ball is drawn at random and the color is noted and then put back inside the box. Then, another ball is drawn at random. Find the probability that: a. both are blue. b. the first is red and the second is yellow. In situation 1, the ball was put back inside the box before getting the second ball. The probability of getting the second ball was the same as the probability of getting the first ball. a. P(blue, blue) 12 141 35 35 1225 b. P(red, yellow) = - 126 35 35 1225 Thus, the two events are independent of each otherSituation 2. Consider a box that contains 14 red balls, 12 blue balls, and 9 yellow balls. Suppose that two balls are drawn one after the other without putting back the first ball Find the probability that a. the first is red and the second is blue. b. both balls are yellow In situation 2. if the ball was not placed back in the box, then drawing the two balls would have been dependent events. In this case, the event of drawing a yellow ball on the second draw is dependent on the event of drawing a yellow ball on the first draw. a. P (red, blue) all , 12 - 168 84 35 34 1190 595 b. P (yellow, yellow) = 2 . & = 72 - 18 1190 595 Example 1: A box contains 7 white marbles and 7 red marbles. What is the probability of drawing 2 white marbles and 1 red marble in succession without replacement? On the first draw, the probability of getting a white marble is -. On the second draw, the probability of getting a white marble is . Then on the third draw, the probability of getting a red marble is- So, P(1 white, 1 white, 1 red) = 2. 6. 1 = 14 13 12 If two events, A and B, are dependent, then the probability of both events occurring is the product of the probability of A and the probability of B after A occurs. In symbols, P(A and B) = P(A) . P(B following A) or Multiplication Rule in Probability Example 2. Suppose you take out two cards from a standard pack of cards one after another, without replacing the first card. What is probability that the first card is the ace of spades, and the second card is a heart? P(1st card is the ace of spades)= If the ace of spaces is drawn first, then there are 51 cards left in the deck, of which 13 are hearts: P(2nd card is a heart | 1st card is the ace of spades)= So, by the multiplication rule of probability, we have: Place of spades, then a heart)= _ , 13 = 13- 1 2652 204 B. EXERCISES: Exercise 1. Directions: Write the letter of the correct answer on your answer sheets. Show your solutions. 1. A bowl contains 15 chips numbered 1 to 15. If a chip is drawn randomly from the bowl, what is the probability that it is a. 7 or 15? A. B. C. D. b, 5 or a number divisible by 3? A. B. PIN c. even or divisible by 3? D. A. B. C. D.d. a number divisible by 3 or divisible by 4? 2 D. 1 B. C. 15 A. 15 15 2. Dario puts 44 marbles in a box in which 14 are red, 12 are blue, and 18 are yellow. If Dario picks one marble at random, what is the probability that he selects a red marble or a yellow marble? C. A. B. D. . Exercise 2: Directions: Read carefully the situation and answer the following: 1. Two cards are drawn in succession from a standard deck of cards with replacement. What is the probability that the first drawn is a diamond, the second is a club? A. B. 4 C. 2. Nick has 4 black pens, 3 blue pens, and 2 red pens in his school bag. Nick randomly picks two pens out of his school bag. What is the probability that Nick chose two black pens, if he replaced the first pen back in his pocket before choosing a second pen? A. - B. 16 C. D. Ta For number 3-5 A bag contains 3 red balls and 2 white balls. A ball was drawn at random and its color was noted down. Without replacement, another ball was drawn. Find the probability that the ball drawn are: 3. both red D. A. B. 7 C. 4. red followed by white C. D. 70 5. both white. A. B. C. D. C. Assessment/Application/Outputs A. Consider the situations below and answer the questions that follow. Write the letter of the correct answer on your answer sheet. 1. A toy box contains 12 toys, 8 stuffed animals, and 3 board games. Maria randomly chooses 2 toys for the child she is babysitting to play with. What is the probability that she chose 2 stuffed animals as the first two choices? A . 23 B. C . 253 D. 506 2. A basket contains 6 apples, 5 bananas, 4 oranges, and 5 guavas. Dominic randomly chooses one piece of fruit, eats it, and chooses another piece of fruit. What is the probability that he chose a banana and then an apple? A. 30 380 B. TO C. D. 38 3. On any given night, the probability that Nick has a cookie for dessert is 10%. The probability that Nick has ice cream for dessert is 50%. The probability that Nick has a cookie or ice cream is 55%. What is the probability that Nick has a cookie and ice cream for dessert? A. 0.05 B. 0.55 C. .60 D. 0.65 4. Sarah tells her mom that there is a 40% chance she will clean her room, a 70% she will do her homework, and a 24% chance she will clean her room and do her homework. What is the probability of Sarah cleaning her room or doing her homework? A. 40% B. 86% C . 90% D. 100%