Math is kinda hard please help me

A. Discussion and Examples Perhaps, you are now familiar with the concept of probability and how it helped us in solving many problems that involved chances. This time, we will further enhance your problem-solving skills in probability related problems as we consider these problems presented in this section. Task 1: A group of Grade 10 students in a certain school were surveyed about their preferred learning modality. It was found that 80% preferred modules (M), 65% preferred online class (O) and 15% did not prefer both. A. Assuming that the survey reflects the actual students of the school, what is the probability that a student selected at random: a. prefers at least one of the two modalities? b. prefers both modalities? B. Draw a Venn diagram representing the result of the survey. Discussion for Task 1: A. The problem is an example of Union and Intersection in probability a. Since 15% prefer neither modality, that means 85% prefer at least one of the two modalities. Thus, the probability that a student elected at random prefers at least one of the modalities is P(M U O) = 0.85. b. Using the General Addition Rule to find the probability that a student selected prefers both modalities, we write: P(M U O) = P(M) + P(O) - P(M no) By substitution, 0.85 = 0.8 +0.65 - P(M n O) P(M n O) = 1.45 -0.85 P(M n 0) = 0.6 Therefore, the probability that a selected student prefers both modalities is 0.60 or 60%. B. The Venn diagram below represents the result of the survey. M O 020 1 0:20 0:05 0.15 You try. What is the probability that a number selected at random from the first 50 positive integers is exactly divisible by 3 or 57 Before you look at the solution below, try answering it by yourself first.Solution: From numbers 1 - 50, each number is as likely to be chosen as any other. Let T = numbers divisble by 3, F numbers divisible by 5 and A = T UF. There are 16 numbers from 1 to 50 divisible by 3. P(T) = 25 There are 10 numbers from 1 to 50 divisible by 5. P(F) = = There are 3 numbers from 1 to 50 divisible by both 3 and 5. P(T n F) = So, P(A) = P(T) + P(F) - P(T ( F ) By substitution, P(TUF) = +1.3- 23 5 50 50 Therefore, the probability of getting a number that is exactly divisible by 3 or 5 is 50 OF 46%. Task 2: Fifty tickets were sold for a car raffle and you bought eight tickets. What is the probability that your tickets will not win the raffle? Discussion for Task 2: The problem falls under the probability of a complement of an event with the formula P(W') = 1 - P(W) where W' = not winning and W = winning and P(W) = = =0.16. By substitution, P(W') = 1-0.16 P(W') = 0.84 Therefore, the probability that your tickets will not win the raffle is 0.84 or 84%. You try . In a survey of 1 000 Filipinos on whether they want to get vaccinated or not, 585 want to get vaccinated and 203 are not sure if they want to get vaccinated. During the survey, what is the probability that a certain respondent does not want to get vaccinated? Before you look at the solution below, try answering it by yourself first. Solution: First, let us get the probability of those who want to get vaccinated and those who are not sure. We say, P(V) = 0.585 and the P(U) = 0.203. Then let's get the P(V u U) = P(V) + P(U) = 0.585 + 0.203 = 0.788. To get the probability of having a certain respondent who doesn't want to get vaccinated, we write P(V UU)' 1 - P(VU U) = 1-0.788 = 0.212. Therefore, the probability of having a certain respondent who doesn't want to get vaccinated is 0.21? or 21.2%. Task 3: How is counting technique used in probability? Let's consider this. A class of 42 grade 10 students is composed of 16 boys and 26 girls. Suppose a team compose of three members is created, what is the probability that the team is an all-girls team? Discussion for Task 3: The event of composing a team of certain number of members is a usual example of a combinationproblem. This time, combination is used to solve a problem of finding the probability. To solve this problem, we start by representing the number of events and the number of the sample space. Number of outcomes in the event = 26Ca = 2 600 Number of outcomes in the sample space - 42Ca = 11,480 Then, we will represent the probability as P( 3 girls team) = 26Ca / 47Ca = =0.2265. 11/480 Therefore, the probability that the team is an all-girls team is 0.2265 or 22.65%. You try. Ten of the 40 marbles in a pouch are blue while the rest are in different colors. If you are going to pick marbles, one at a time, without replacement, what is the probability that all the marbles you pick are not blue? Before you look at the solution below, try answering it by yourself first. Solution: Since we are looking for the probability of picking marbles that are not blue, we know that there are 30 of them in the pouch. Now, we can represent the number of outcomes in the event and the number of outcomes in the sample space. Number of outcomes in the event = 30Cs = 142, 506 Number of outcomes in the sample space = 40Cs = 658, 008 The probability is P(5 not blue marbles) = 30Cs / 40Cs = = 0.2166 658,008 Therefore, the probability that all the marbles you pick are not blue is 0.2166or 21.66%. Task 4: Study the event and figure out what probability is used here. A cage contains 5 female chicks and 4 male chicks. You randomly picked 3 chicks one at a time without putting it back. What is the probablety that you can pick 3 male chicks? Discussion for Task 3: The problem presented will apply probability of dependent events to solve it. The formula for the probability of dependent events is P(M, , Mz , Ma) = P(M,) x P(My following M,) x PIM, following (M2 following M,)). Since there are 4 male chicks out of the 9 chicks, then way say that, P(M1) = =, P(M2 following M,) =-, P(M, following (Mz following M,) = = So, P(M, , Mz, M3) = 3 x x 2= 24 =0.0476 304 Therefore, the probability of picking 3 male chicks one at a time is 0.0476 or 4.76%. You try. This time you try probability of independent events. Two dice are rolled simultaneously. What is the probability that the first die shows a number less than 5 and the second die shows an even number? Before you look at the solution below, try answering it by yourself first. Solution: Since there are two dice, we say that the events are independent. Let P(D,) = = = = and P(D2) = = = =P(D, & D,) = P(D, ) X P(D,) = = x - Therefore, the probability that the first die shows a number less than 5 and the second die shows an even number is - or 33.33%. B. Exercises Exercise 1 Directions: Consider the probability problems below and answer the questions asked for each item. Show your complete and organize solution on a separate sheet of paper. 1. There are 5 yellow 7 orange balls. Two balls are selected one by one without replacement. Find the probability that first is yellow and second is orange. 2. What is the probability of selecting a number divisible by 4 from a set of 6 consecutive even numbers starting from number 6? 3. A number is chosen from the first 50 natural numbers. What is the probability that the number is odd or divisible by 57 4. A bag contains 100 marbles, 56 red ones and 44 blue ones. What is the probability of picking a red marble after a blue marble had been picked? 5. Find the probability of getting 3 heads when 6 coins are tossed. Exercise 2 Directions: Consider the probability problems below and answer the questions asked for each item. Show your complete and organize solution on a separate sheet of paper. 1. A box contains 30 blue beads, 60 red beads, and 40 yellow beads. If a bead is picked at random from the box, what is the probability that the bead picked is: a. a yellow? b. a blue or red? c. not a blue? 2. Twenty tiles numbered 1 to 20 are placed in a pouch. If a tile is randomly picked from the pouch, what is the probability that it is: a. 4 or 8? b. 7 or a number divisible by 2? c. both even and divisible by 3? 3. A two-digit number is written at random. A. Determine the number: a. of odd numbers. b. of numbers larger than 75. c. of numbers multiple of 5. d. of two digit numbers. B. Determine that the probability that the number will be: a, an odd number. b. larger than 75. c, a multiple of 5. 4. In a standard deck of 52 cards, find the probability of obtaining: a. a heart. b. a spade.c. a red card d. 24. 5. in a Grade 10 class of 42 students, 8 joined Math Club only, 12 joined Science Club only, and 20 joined both In Math and Science Clubs. If a student is selected at random, what is the probability that the selected student is: a. joining Math Club? b. joining Science Club? not joining any club? C. Assessment/Application/Outputs Directions: Encircle the letter of your answer for multiple choice items. Write your answer for those items that don't have choices. Show your solution on a clean sheet of paper to items that need solution. 1. What is the probability of getting a sum of 9 when two dice are thrown? a. b . = C. 36 d. = 2. What is the probability as getting at least one '5' when three dice are rolled together? 125 d. 213 a. 216 b. C. 216 3. A box contains 5 blue chips and 8 green chips. A chip is removed and replaced by two of the other color and then a second chip is drawn. What is the probability that the second chip is a green? a. $3 91 b. 29 91 C. 97 d. 31 4. In a class in which all students joined at least one club, 60% of students join volleyball or basketball and 10% joined both club. If there is also 60% that do not join volleyball, what is the probability that a student chosen at random from the class joined neither volleyball nor basketball? a. 3 b . 3 C. 3 d. 5. In a certain town of Cebu, 40% of the population is male, 25% are children and 15% are male children. A person is chosen at random, if he is a male, what is the probability that he is also a child? a. 50% b. 37.5% C. 62.5% d. 75% 6. There are 100 students in Grade 10, of whom 40 are boys, 30 wear glasses, and 15 are boys who wear glasses. If one Grade 10 student is randomly selected, what is the probability that the student will be a girl who wears glasses? a. 70% b. 60% c. 15% d. 45% 7. On a book shelf, there are 60 fiction books and 20 nonfiction books. Ron chooses a book at random off the book shelf and brings it. Shortly after, Rav chooses another book at random. What is the probability that Rav selected a fiction book? a. 75% b. 65% c. 85% d. 55% 8. Two dice are rolled while two coins are tossed. What is the probability that the two dice will show odd numbers while the two coins show heads? a. b. = C. = d. For Items 9 and 10 The blood groups of 150 people are distributed as follows: 35 have type A blood, 50 have type B blood, 55 have type O blood and 10 have type AB blood. . If a person from this group is selected at random, what is the probability that this person has either type AB blood or type B blood? a. 40% b. 2.22% c. 42.22% d. 37.78% 10. What is the probability that two persons selected both have type O blood? a. 36.67% b. 25% c. 55% d. 13.29% 11. Two dice are rolled. Find the probability that the dice show a sum of 5 or 7. Express your answer asdecimal or percent. For Items 12 and 13. A box of 32 bulbs contains 25% defective bulbs. A bulb is drawn at random from the lot and is never placed back. 12. What is the probability that this bulb is not defective? Express your answer as a fraction. 13. Find the probability that the first bulb is not defective and the second bulb is defective. Express your answer in four decimal places. 14. It is said that the Summative Test will fall within the weekdays of the first three weeks of March this year. What is the probability that it will not fall on a Monday? Show your answer in fraction form. 15. A coin is tossed, a die is rolled, and a card from a standard deck of 52 cards is drawn. What is the probability that a coin shows a tail, a die shows a number less than 4, and a card drawn is not a heart? Show your answer in fraction form. D. Suggested Enrichment/Reinforcement Activity/ies 1. There are many instances in our lives that we need to choose wherein our choices changed our lives. As a Gra de 10 student, write an instance in your life where you make a decision based on the concept of probability. Explain why you use probability before you made that decision. Share also the outcome of that decision. is it favourable or not favourable to you? 2. Construct a probability problem based on the things that you see around you and make an elaborate explanation of your work by labelling the type of probability that you construct and by showing a detailed solution of your work