PPS0034: Introduction to Probability and Statistics Assignment, MMU, Malaysia A committee of 6 people is to be chosen from a group consisting of 7 men and 8 women. Find the probability that the committee

Question 1

• A committee of 6 people is to be chosen from a group consisting of 7 men and 8 women. Find the probability that the committee must consist of at least 3 women and at least 2 men.
• An access code consists of a letter followed by 4 digits. If any letter can be used, the first digit cannot be zero and each digit can be repeated, find the number of possible access codes.
• During the Student Representative Council (SRC) committee election, each member must elect a committee of five people from candidates consisting of six boys and five girls. Find the probability that the committee will have two girls and three boys if a certain girl must be on the committee.
• A group of kindergarten students which consists of eight boys and six girls, four kindergarten students are selected to follow a trip to Zoo Negara so that at least three boys are there on the trip. In how many ways can it be done?
• In a Mathematics course, the students are required to complete four projects. If there are ten different projects to choose from, how many ways can a student choose the four projects?
  i) if there are no restrictions?
  ii) if two of the projects are compulsory for all students?
• How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7, and 9, which are
  divisible by 5 and none of the digits is repeated?
• A family has just moved to a new city and requires the services of both an obstetrician and a pediatrician. There are two easily accessible medical clinics, each having two obstetricians and three pediatricians. The family will obtain maximum health insurance benefits by joining a clinic and selecting both doctors from that clinic. In how many ways can this be done?

• The local Family Restaurant has a daily breakfast special in which the customer may choose one item from each of the following groups:
  
  • i. How many different breakfast specials are possible?
    ii. How many different breakfast specials without meat are possible?
  • There are fourteen juniors and twenty-three seniors in the MMU Community
    Club. The club is to send four representatives to the MMU charity event
    i. How many different ways are there to select a group of four students to
    attend the charity event?
    ii. If the members of the club decide to send two juniors and two seniors,
    what is the probability of different groupings?
  • The football team has 20 players. There are always 11 players on the field. How many different groups of players can be on the field at the same time?
  • There are six men and five women in a small office. The manager wants to form a group of 4 workers. What is the probability of different groups can be formed if at least 3 men must be in the group?
  •  How many passwords can be formed using 6 digits where the first digit must be letters with case sensitive and the last four digits must be numbers?
  • A cricket eleven can be chosen out of a batch of 18 players.
    i) How many ways if there is no restriction on the selection?
    ii) How many ways is a particular player always chosen?
    iii) What is the probability that a particular player is never chosen?
  • There are 11 freshmen, 5 sophomores, and 10 juniors in a chess club. A group of 9 students will be chosen to compete in a competition.
    i) How many combinations of students are possible if the group is to consist of an equal number of freshmen, sophomores, and juniors?
    ii) What is the probability that the combinations of students are possible if the group is to consist of all members of the same class?
  • A bucket contains the following marbles: 4 red, 3 blue, 4 green, and 3 yellow. Each marble is labeled with a number so it can be distinguished. 4 marbles are chosen at random.
    i) What is the probability that each one is a different color?
    ii) What is the probability that all must be the same color?
    iii) What is the probability that at least two red and one yellow marble are chosen?
  • If 5 products are picked at random from a shelf at a supermarket containing 5 cans of tuna, 10 cans of sardine, and 15 cans of baked beans,
    a) in how many ways can the product be chosen if there are no restrictions?
    b) what is the probability that 3 cans of tuna and 2 cans of baked beans are selected?
    c) what is the probability that at least 4 cans of sardine are selected?

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  Question 2

  • The probability that a planted radish seed germinates is 0.69. A gardener plants nineteen seeds. Let X denote the number of radish seeds that successfully germinate.
    i) What is the probability of 6 to 8 radish seeds that successfully germinate?
    ii) What is the probability of not more than 7 radish seeds successfully germinating?
    iii) What is the probability of 2 or fewer radish seeds that do NOT successfully germinate?
    iv) What is the average number of seeds the gardener could expect to germinate?
  • The deals cracked by an agent per day is a Poisson random variable with a standard deviation of 3. Given that each day is independent of the other day, find the probability of
    i) exactly 14 deals cracked per 2 days?
    ii) getting less than 3 deals cracked on the first day and at most 10 deals to be cracked the next day.
  • Suppose that the weights of bags of potato chips coming from a factory follow a normal distribution with a mean of 12.8 ounces and a standard deviation of 0.6 ounces.
    i) What is the probability of the bags weighing between 10.5 ounces and 13.7 ounces?
    ii) If the manufacturer wants to keep the mean at 12.8 ounces but adjust the standard deviation so that only 1% of the bags weigh less than 12 ounces, how small does he/she need to make that standard deviation?
  • On average, airline A has 6 mishandled bags per 1000 passengers.
    i. What is the probability that for the next 1000 passengers, the airline will have no mishandled bags?
    ii. What is the probability that for the next 1500 passengers, the airline will have at least seven mishandled bags?
  • Suppose that five friends made five orders in a restaurant, which filled approximately 67% of its orders correctly in the last month.
    i. What is the probability that all five orders will be filled correctly?
    ii. What is the probability that none of the five orders will be filled correctly?
    iii. What is the probability that two to four orders will be filled correctly?
  • A study found that consumers spend an average of RM21 per week in cash without being aware of where it goes. Assume that the amount of cash spent is normally distributed and that the standard deviation is RM5.
    i. What is the probability that a randomly selected person will spend between RM7 and RM19?
    ii. Between what two values will the middle 95% of the amounts of cash spent fall?

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  Question 3

  • By some estimates, 32% of Malaysian have no health insurance. Let X denote the number of Malaysian in the sample with no health insurance. Using the Binomial distribution table,
    i) Find the probability that exactly 3 of the 15 sampled have no health insurance.
    ii) Find the probability that at least 12 of the 20 sampled have no health insurance.
    iii) Find the probability that between 5 and 13 of the 20 sampled have health insurance.
  • A shop sells five pieces of shirts every day, from 10 am to 10 pm.
    i) What is the probability of selling more than three shirts today?
    ii) What is the probability of selling at most thirty-one shirts from Monday to
    Thursday?
    iii) What is the probability of selling three to eight shirts for 9 hours?
  • An automobile manufacturer introduces a new model that averages 27 miles per gallon when driven in the city. A person who plans to purchase one of these new cars wrote to the manufacturer for the details of the tests and found out that the standard deviation is 3 miles per gallon. Assume that in-city mileage is approximately normally distributed.
    i) What is the probability that the person will purchase a car that averages less than 20 miles per gallon for in-city driving?
    ii) What is the probability that the person will purchase a car that averages between 25 and 29 miles per gallon for in-city driving?
  • A toll-free phone number is available from 9 a.m. to 9 p.m. for customers to register complaints about a product purchased from a certain company. Past history indicates that an average of 1.3 calls are received per minute.
    i. What is the probability that, during a 1-minute period, three or more phone calls will be received?
    ii. What is the probability that, during a 5-minute period, from four to eight phone calls will be received?
    iii. Find the mean number of phone calls received during a 1-hour period.
  • In a survey conducted by Human Resource Management, 68% of workers said that employers have the right to monitor their telephone usage. Suppose that a random sample of 20 workers is selected and they are asked if employers have the right to monitor telephone usage. What is the probability that
    i. 10 or fewer of the workers agree?
    ii. between 9 to 15 workers agree?
    iii. less than 4 workers not agree?
  • An orange juice producer buys all his oranges from a large orange grove. The amount of juice squeezed from each of these oranges is approximately normally distributed, with a mean of 4.70 ounces and a standard deviation of 0.30 ounces.
    i. What is the probability that a randomly selected orange will contain between 4.50 and 5.00 ounces of juice?
    ii. 77 % of the oranges will contain at least how many ounces of juice?

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