Use the Gaussian elimination to determine the solution set to the given system
Example 2: The following gambling game, known as the wheel of fortune (or chuck-a-luck), is quite popular at
many carnivals and gambling casinos: A player bets on one of the numbers 1 through 6. Three dice are then rolled,
and if the number bet by the player appears i times, i = 1, 2, 3, then the player wins i units; if the number bet by
the player does not appear on any of the dice, then the player loses 1 unit. Is this game fair to the player?
(Actually, the game is played by spinning a wheel that comes to rest on a slot labeled by three of the numbers 1
through 6, but this variant is mathematically equivalent to the dice version.)
Example 3: A communication system consists of n components, each of which will, independently, function with
probability p. The total system will be able to operate effectively if at least one-half of its components function.
For what values of p is a 5-component system more likely to operate effectively than a 3-component system?
Example 6: It is known that screws produced by a certain company will be defective with probability .01,
independently of each other. The company sells the screws in packages of 10 and offers a money-back guarantee
that at most 1 of the 10 screws is defective. What proportion of packages sold must the company replace?
Example 7: An airline knows that 5% of the people making reservations on a certain flight will not show up.
Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability
that there will be a seat available for every passenger who shows up.
A method for measuring the pH level of a solution yields a measurement value that is normally distributed with a
mean equal to the actual pH of the solution and with a standard deviation equal to .05. An environmental
pollution scientist claims that two different solutions come from the same source. If this were so, then the pH
level of the solutions would be equal. To test the plausibility of this claim, 10 independent measurements were
made of the pH level for both solutions, with the following data resulting.
Example 2: The following gambling game, known as the wheel of fortune (or chuck-a-luck), is quite popular at
many carnivals and gambling casinos: A player bets on one of the numbers 1 through 6. Three dice are then rolled,
and if the number bet by the player appears i times, i = 1, 2, 3, then the player wins i units; if the number bet by
the player does not appear on any of the dice, then the player loses 1 unit. Is this game fair to the player?
(Actually, the game is played by spinning a wheel that comes to rest on a slot labeled by three of the numbers 1
through 6, but this variant is mathematically equivalent to the dice version.)
Example 3: A communication system consists of n components, each of which will, independently, function with
probability p. The total system will be able to operate effectively if at least one-half of its components function.
For what values of p is a 5-component system more likely to operate effectively than a 3-component system?
Example 6: It is known that screws produced by a certain company will be defective with probability .01,
independently of each other. The company sells the screws in packages of 10 and offers a money-back guarantee
that at most 1 of the 10 screws is defective. What proportion of packages sold must the company replace?
Example 7: An airline knows that 5% of the people making reservations on a certain flight will not show up.
Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability
that there will be a seat available for every passenger who shows up.
A method for measuring the pH level of a solution yields a measurement value that is normally distributed with a
mean equal to the actual pH of the solution and with a standard deviation equal to .05. An environmental
pollution scientist claims that two different solutions come from the same source. If this were so, then the pH
level of the solutions would be equal. To test the plausibility of this claim, 10 independent measurements were
made of the pH level for both solutions, with the following data resulting.
