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Baseball Team Problem 2: Nine people try out for the nine positions on a baseball team a. In how many different ways could the positions be filled if there are no restrictions on who plays which position? b. In how many different ways could the positions be filled if Fred must be the pitcher but the other eight people can take any of the remaining eight positions? c. If the positions are selected at random, what is the probability that Fred will be the pitcher? d. What is the probability in part c expressed as a percent?

Spaceship Problem 2: Complex systems such as spaceships have many components. Unless the system has backup components, the failure of any one component could cause the entire system to fail. Suppose a spaceship has 1000 such vital components and is designed without backups. a. If each component is \(99.9 \%\) reliable, what is the probability that all 1000 components work and the spaceship does not fail? Does the result surprise you? b. What is the minimum reliability needed for each component to ensure that there is a \(90 \%\) probability that all 1000 components will work?

Soccer Team Problem 1: Eleven girls try out for the 11 positions on a soccer team In how many different ways could the 11 positions be filled if there are no restrictions on who plays which position? b. In how many different ways could the positions be filled if Mabel must be the goalkeeper? c. If the positions are selected at random, what is the probability that Mabel will be the goalkeeper? d. What is the probability in part c expressed as a percent?

Measles and Chicken Pox Problem: Suppose that in any one year a child has a 0.12 probability of catching measles and a 0.2 probability of catching chicken pox. a. If these events are independent of each other, what is the probability that a child will get both diseases in a given year? b. Suppose statistics show that the probability for getting measles and then chicken pox in the same year is 0.006 i. Calculate the probability of getting both diseases. ii. What is the probability of getting chicken pox and then measles in the same year? c. Based on the given probabilities and your answers to part b, what could you conclude about the effects of the two diseases on each other?

Airplane Engine Problem 1: One reason airplanes are designed with more than one engine is to increase the planes' reliability. Usually a twin-engine plane can make it to an airport on just one engine should the other engine fail during flight. Suppose that for a twin-engine plane, the probability that any one engine will fail during a given flight is \(3 \%\) a. If the engines operate independently, what is the probability that both engines fail during a flight? b. Suppose flight records indicate that the probability that both engines will fail during a given flight is actually \(0.6 \% .\) What is the probability that the second engine fails after the first has already failed? c. Based on your answer to part \(b\), do the engines actually operate independently? Explain.

Seating Problem: There are 10 students in a class and 10 chairs, numbered 1 through 10 a. In how many different ways could a student be selected to occupy chair \(1 ?\) b. After someone is seated in chair \(1,\) how many different ways are there of seating someone in chair \(2 ?\) c. In how many different ways could chairs 1 and 2 be filled? d. If two of the students are sitting in chairs 1 and \(2,\) in how many different ways could chair 3 be filled? e. In how many different ways could chairs \(1,2,\) and 3 be filled? f. In how many different ways could all 10 chairs be filled? Surprising?!

Eight children line up for a fire drill (Figure \(9-4 b)\) a. How many possible arrangements are there? b. In how many of these arrangements are Calvin and Phoebe next to each other? (Clue: Arrange seven things- the Calvin and Phoebe pair and the other six children. Then arrange Calvin and Phoebe. \()\) c. If the eight students line up at random, what is the probability that Calvin and Phoebe will be next to each other?

Twelve people apply to go on a biology field trip, but there is room in the car for only five of them In how many different ways could the group of five making the trip be chosen? How can you tell that a number of combinations is being asked for, not a number of permutations?

Permutations with Repeated Elements-Problems 15 and 16: The word CARRIER has seven letters. But there are fewer than \(7 !\) permutations, because in any arrangement of these seven letters the three \(R\) 's are interchangeable. If these \(R\) 's were distinguishable, there would be \(3 !\), or \(6,\) ways of arranging them. This implies that only \(\frac{1}{6}\) (that is, \(\frac{1}{3 !}\) ) of the \(7 !\) permutations are actually different. So the number of permutations is $$ \frac{7 !}{3 !}=840 $$ There are four \(I^{\prime}\) 's, four \(S\) 's, and two \(P\) 's in the word MISSISSIPPI, so the number of different permutations of its letters is $$ \frac{111}{4 ! 4 ! 2 !}=34,650 $$ Nine pennies are lying on a table. Five show heads and four show tails. In how many different ways, such as "HHTHTTHHT," could the coins be lined up if you consider all the heads to be identical and all the tails to be identical?

Telephone Number Problem: When 10 -digit telephone numbers were introduced into the United States and Canada in the 1960 s, certain restrictions were placed on the groups of numbers: Area Code: 3 digits; the first must not be 0 or \(1,\) and the second must be 0 or 1 Exchange Code: 3 digits; the first and second must not be 0 or 1 Line Number: 4 digits; at least one must not be 0 a. Find the possible numbers of area codes, exchange codes, and line numbers. b. How many valid numbers could there be under this numbering scheme? c. How many 10-digit numbers could be made if there were no restrictions on the three groups of numbers? d. What is the probability that a 10 -digit number dialed at random would be a valid number under the original restrictions? e. The total population of the United States and Canada is currently about 300 million. In view of the fact that there are now area codes and exchange codes that do not conform to the original restrictions, what assumption can you make about the number of telephones per person in the United States and Canada?