Week 2 Practice Problems Part 1: Definition of Probability, Finite Simple Sample Spaces 1. One ball is to be selected from a box containing red, white, blue, yellow, and green balls. If the probability that the ball selected will be red is 1/5 and the probability that the ball selected will be white is 2/5, what is the probability that the ball will be blue, yellow, or green? 2. Consider two events and such that and . Determine the value of ∩
for each of the following conditions: a)
and are disjoint. b)
⊂ c)
∩
3. Consider two events and , where 0.4 and 0.7. Determine the maximum and minimum possible values of ∩ and the conditions under which each of these occur. 4. A box contains 24 light bulbs, of which 4 are defective. A person selects 4 bulbs, without replacement. a) What is the probability that all 4 selected bulbs will be defective? b) What is the probability that at least 2 of the bulbs selected are defective? 5. A box contains 24 light bulbs, of which 4 are defective. If one person selects 10 bulbs from the box in a random manner and a second person selects the remaining 14 bulbs, what is the probability that all four defective bulbs will be obtained by the same person? 6. A box in a certain supply room contains three types of bulbs (40 Watt, 60 Watt, and 75 Watt). There are four 40‐W light bulbs, five 60‐W bulbs, and six 75‐W bulbs. Suppose that three bulbs are randomly selected. a) What is the probability that exactly two of the selected bulbs are 75‐W? b) What is the probability that all three of the selected bulbs are the same type? c) What is the probability that exactly one of each type of bulb is selected? 7. Suppose that letters are placed at random into envelopes. What is the probability that exactly 1 of the letters will be placed into the correct envelopes? 8. Suppose that 3 letters are placed at random in 3 envelopes. What is the probability that exactly 1 of the letters will be placed into the correct envelopes? 9. A store has 80 modems in its inventory; 30 came from source A and 50 came from source B. Of the modems from source A, 20% are defective. Of the modems from source B, 8 % are defective. Calculate the probability that exactly two out of a random sample of five modems from the store’s inventory are defective. Part 2: Conditional Probability, Independence, Law of Total Probability, & Bayes’ Theorem 1. If and are disjoint sets with 0, what is | ? 2. Assume that and are independent. Prove that and are independent. 3. If and are independent events with 1, what is the value of |
? 4. Urn 1 contains 2 white balls and 2 black balls. Urn 2 contains 3 white balls and 2 black balls. An urn is chosen at random, and a ball is randomly selected from that urn. Find the probability that the ball chosen is white. 5. A fair six‐sided die is tossed. Define the following events: 
is the event that the number rolled is even 
is the event that the number tossed is 3 or less. 
is the event that the number tossed is a 1 or a 2. a) Describe the sample space and the given events as sets. b) Are and independent? What about and ? What about and ? 6. An actuary studying the insurance preferences of automobile owners makes the following conclusions:  An automobile owner is twice as likely to purchase collision coverage as disability coverage.  The event that an automobile owner purchases collision coverage is independent of the event that he/she purchases disability coverage.  The probability that an automobile owner purchases both collision and disability coverage is 0.15. What is the probability that an automobile owner purchases neither collision nor disability coverage? 7. Two students and are both registered for a certain course. Assume that student attends class 80% of the time, student attends class 60% of the time, and the absences of the two students are independent. a) What is the probability that at least one of the two students will be in class on a given day? b) If at least one of the two students is in class on a given day, what is the probability that is in class that day? 8. The probability that a randomly chosen male has a circulation problem is 0.25. It is found that males who have a circulation problem are twice as likely to be smokers as those who do not have a circulation problem. What is the conditional probability that a male has a circulation problem given that he is a smoker? 9. An auto insurance company insures drivers of all ages. An actuary compiled the following statistics on the company’s insured drivers: Age of Driver Probability of Accident Portion of Companies Insured Drivers 16‐20 0.06 0.08 21‐30 0.03 0.15 31‐65 0.02 0.49 66‐99 0.04 0.28 A randomly selected driver that the company insured has an accident. Calculate the probability that the driver was age 16‐20. 10. A doctor is studying the relationship between blood pressure and heartbeat abnormalities in her patients. She tests a random sample of her patients and notes their blood pressures (High, Low, or Normal) and their heartbeats (Regular or Irregular). She finds that:  14% have high blood pressure.  22% have low blood pressure.  15% have an irregular heartbeat.  Of those with an irregular heartbeat, one‐third have high blood pressure.  Of those with normal blood pressure, one‐eighth have an irregular heartbeat. What portion of the patients selected have a regular heartbeat and low blood pressure?