6.8: Chapter 6 Review
SECTION 6.8 PROBLEM SET: CHAPTER REVIEW
-
Two dice are rolled. Find the probability that the sum of the dice is
- four
- five
-
A jar contains 3 red, 4 white, and 5 blue marbles. If a marble is chosen at random, find the following probabilities:
- P(red or blue)
- P(not blue)
-
A card is drawn from a standard deck. Find the following probabilities:
- P(a jack or a king)
- P(a jack or a spade)
-
A basket contains 3 red and 2 yellow apples. Two apples are chosen at random. Find the following probabilities:
- P(one red, one yellow)
- P(at least one red)
-
A basket contains 4 red, 3 white, and 3 blue marbles. Three marbles are chosen at random. Find the following probabilities:
- P(two red, one white)
- P(first red, second white, third blue)
- P(at least one red)
- P(none red)
-
Given a family of four children. Find the following probabilities:
- P(All boys)
- P(1 boy and 3 girls)
-
Consider a family of three children. Find the following:
- P(children of both sexes | first born is a boy)
- P(all girls | children of both sexes)
- Mrs. Rossetti is flying from San Francisco to New York. On her way to the San Francisco Airport she encounters heavy traffic and determines that there is a 20% chance that she will be late to the airport and will miss her flight. Even if she makes her flight, there is a 10% chance that she will miss her connecting flight at Chicago. What is the probability that she will make it to New York as scheduled?
- At a college, twenty percent of the students take history, thirty percent take math, and ten percent take both. What percent of the students take at least one of these two courses?
- In a T-maze, a mouse may run to the right (R) or may run to the left (L). A mouse goes up the maze three times, and events E and F are described as follows: \[ \text{ E: Runs to the right on the first trial } \quad \text{ F: Runs to the left two consecutive times } \nonumber \] Determine whether the events E and F are independent.
-
A college has found that 20% of its students take advanced math courses, 40% take advanced English courses and 15% take both advanced math and advanced English courses. If a student is selected at random, what is the probability that
- he is taking English given that he is taking math?
- he is taking math or English?
- If there are 35 students in a class, what is the probability that at least two have the same birthday?
- A student feels that her probability of passing accounting is .62, of passing mathematics is .45, and her passing accounting or mathematics is .85. Find the probability that the student passes both accounting and math.
- There are nine judges on the U. S. Supreme Court. Suppose that five are conservative and four are liberal. This year the court will act on six major cases. What is the probability that out of six cases the court will favor the conservatives in at least four?
-
Five cards are drawn from a deck. Find the probability of obtaining
- four cards of a single suit
- two cards of one suit, two of another suit, and one from the remaining
- a pair(e.g. two aces and three other cards)
- a straight flush(five in a row of a single suit but not a royal flush)
- The following table shows a distribution of drink preferences by gender.
|
Coke(C) |
Pepsi(P) |
Seven Up(S) |
TOTALS |
|
|
Male(M) |
60 |
50 |
22 |
132 |
|
Female(F) |
50 |
40 |
18 |
108 |
|
TOTALS |
110 |
90 |
40 |
240 |
The events M, F, C, P and S are defined as Male, Female, Coca Cola, Pepsi, and Seven Up, respectively. Find the following:
- P(F | S)
- P( P | F)
- P(C | M)
- P(M | P \(\cup\) C)
- Are the events F and S mutually exclusive?
- Are the events F and S independent?
- At a clothing outlet 20% of the clothes are irregular, 10% have at least a button missing and 4% are both irregular and have a button missing. If Martha found a dress that has a button missing, what is the probability that it is irregular?
-
A trade delegation consists of four Americans, three Japanese and two Germans. Three people are chosen at random. Find the following probabilities:
- P(two Americans and one Japanese)
- P(at least one American)
- P(One of each nationality)
- P(no German)
- A coin is tossed three times, and the events E and F are as follows. \[ \text { E: It shows a head on the first toss } \quad \text { F: Never turns up a tail } \nonumber\] Are the events E and F independent?
- If \(P(E) = .6\) and \(P(F) = .4\) and E and F are mutually exclusive, find \(P\)(E and F).
- If \(P(E)=.5\) and \(P(F)=.3\) and E and F are independent, find \(P(E \cup F)\).
- If \(P(F)=.9\) and \(P(E | F)=.36\) and E and F are independent, find \(P(E)\).
- If \(P(E)=.4\) and \(P\)(E or F) =.9 and E and F are independent, find \(P(F)\).
- If \(P(E) = .4\) and \(P(F | E) = .5\), find \(P\)(E and F).
- If \(P(E) = .6\) and \(P\)(E and F) = .3, find \(P(F | E)\).
- If \(P(E ) = .3\) and \(P(F) = .4\) and E and F are independent, find \(P(E | F)\).
-
A coin is tossed five times. Find the following
- P(2 heads and 3 tails)
- P(at least 4 tails)
- A dandruff shampoo helps 80% of the people who use it. If 10 people apply this shampoo to their hair, what is the probability that 6 will be dandruff free?
- A baseball player has a .250 batting average. What is the probability that he will have 2 hits in 4 times at bat?
- Suppose that 60% of the voters in California intend to vote Democratic in the next election. If we choose five people at random, what is the probability that at least four will vote Democratic?
- A basketball player has a .70 chance of sinking a basket on a free throw. What is the probability that he will sink at least 4 baskets in six shots?
- During an archery competition, Stan has a 0.8 chance of hitting a target. If he shoots three times, what is the probability that he will hit the target all three times?
- A company finds that one out of four new applicants overstate their work experience. If ten people apply for a job at this company, what is the probability that at most two will overstate their work experience?
- A missile has a 70% chance of hitting a target. How many missiles should be fired to make sure that the target is destroyed with a probability of .99 or more?
-
Jar I contains 4 red and 5 white marbles, and Jar II contains 2 red and 4 white marbles. A jar is picked at random and a marble is drawn. Draw a tree diagram and find,
- P(Marble is red)
- P(It is white given that it came from Jar II)
- P(It came from Jar II knowing that the marble drawn is white)
- Suppose a test is given to determine if a person is infected with HIV. If a person is infected with HIV, the test will detect it in 90% of the cases; and if the person is not infected with HIV, the test will show a positive result 3% of the time. If we assume that 2% of the population is actually infected with HIV, what is the probability that a person obtaining a positive result is actually infected with HIV?
- A car dealer’s inventory consists of 70% cars and 30% trucks. 20% of the cars and 10% of the trucks are used vehicles. If a vehicle chosen at random is used, find the probability that it is a car.
-
Two machines make all the products in a factory, with the first machine making 30% of the products and the second 70%. The first machine makes defective products 3% of the time and the second machine 5% of the time.
- Overall what percent of the products made are defective?
- If a defective product is found, what is the probability that it was made on the second machine?
- If it was made on the second machine, what is the probability that it is defective?
-
An instructor in a finite math course estimates that a student who does his homework has a 90% of chance of passing the course, while a student who does not do the homework has only a 20% chance of passing the course. It has been determined that 60% of the students in a large class do their homework.
- What percent of all the students will pass?
- If a student passes, what is the probability that he did the homework?
-
Cars are produced at three factories. Factory I produces 10% of the cars and it is known that 2% are defective. Factory II produces 20% of the cars and 3% are defective. Factory III produces 70% of the cars and 4% of those are defective. A car is chosen at random. Find the following probabilities:
- P(The car is defective)
- P(The car came from Factory III | the car is defective)
- A stock has a 50% chance of a 10% gain, a 30% chance of no gain, and otherwise it will lose 8%. Find the expected return.
- A game involves rolling a pair of dice. One receives the sum of the face value of both dice in dollars. How much should one be willing to pay to roll the dice to make the game fair?
- A roulette wheel consists of numbers 1 through 36, 0, and 00. If the wheel shows an odd number you win a dollar, otherwise you lose a dollar. If you play the game ten times, what is your expectation?
- A student takes a 100-question multiple-choice exam in which there are four choices to each question. If the student is just guessing the answers, what score can he expect?
- Mr. Shaw invests 50% of his money in stocks, 30% in mutual funds, and the remaining 20% in bonds. If the annual yield from stocks is 10%, from mutual funds 12%, and from bonds 7%, what percent return can Mr. Shaw expect on his money?
- An insurance company is planning to insure a group of surgeons against medical malpractice. Its research shows that two surgeons in every fifteen are involved in a medical malpractice suit each year where the average award to the victim is $450,000. How much minimum annual premium should the insurance company charge each doctor?
- In an evening finite math class of 30 students, it was discovered that 5 students were of age 20, 8 students were about 25 years old, 10 students were close to 30, 4 students were 35, 2 students were 40 and one student 55. What is the average age of a student in this class?
- Jar I contains 4 marbles of which one is red, and Jar II contains 6 marbles of which 3 are red. Katy selects a jar and then chooses a marble. If the marble is red, she gets paid 3 dollars, otherwise she loses a dollar. If she plays this game ten times, what is her expected payoff?
- Jar I contains 1 red and 3 white, and Jar II contains 2 red and 3 white marbles. A marble is drawn from Jar I and put in Jar II. Now if one marble is drawn from Jar II, what is the probability that it is a red marble?
- Let us suppose there are three traffic lights between your house and the school. The chance of finding the first light green is 60%, the second 50%, and the third 30%. What is the probability that on your way to school, you will find at least two lights green?
- Sonya has just earned her law degree and is planning to take the bar exam. If her chance of passing the bar exam is 65% on each try, what is the probability that she will pass the exam in at least three tries?
- Every time a particular baseball player is at bat, his probability of getting a hit is .3, his probability of walking is .1, and his probability of being struck out is .4. If he is at bat three times, what is the probability that he will get two hits and one walk?
- Jar I contains 4 marbles of which none are red, and Jar II contains 6 marbles of which 4 are red. Juan first chooses a jar and then from it he chooses a marble. After the chosen marble is replaced, Mary repeats the same experiment. What is the probability that at least one of them chooses a red marble?
- Andre and Pete are two tennis players with equal ability. Andre makes the following offer to Pete: We will not play more than four games, and anytime I win more games than you, I am declared a winner and we stop. Draw a tree diagram and determine Andre's probability of winning.