Calculus and Above

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Task Background: A large...

Task Background: A large part—but not all—of this assignment concerns a situation in which you are involved in the programming of electronic games. To carry out these tasks effectively, a working knowledge of sets, relations, combinatorics, and probability is necessary.

Part I: Set Theory

Look up a roulette wheel diagram. The following sets are defined:

A = the set of red numbers

B = the set of black numbers

C = the set of green numbers

D = the set of even numbers

E = the set of odd numbers

F = {1,2,3,4,5,6,7,8,9,10,11,12}

From these, determine each of the following:

A∪B

A∩D

B∩C

C∪E

B∩F

E∩F

Part II: Relations and Functions

The implementation of the program that runs the game involves testing. One of the necessary tests is to see if the simulated spins are random. Create an n-ary relation, in table form, that depicts possible results of 10 trials of the game. Include the following results of the game:

Number

Color

Odd or even (note: 0 and 00 are considered neither even nor odd.)

Also include a primary key. What is the value of n in this n-ary relation?

Part III: Graphs and Trees

Create a tree that models the following scenario. A player decides to play a maximum of 4 times, betting on red each time. The player will quit after losing twice. In the tree, any possible last plays will be an ending point of the tree. Branches of the tree should indicate the winning or losing, and how that affects whether a new play is made.

Part IV: Combinatorics and Probability

In the roulette game, what is the probability of an outcome of:

Any odd number

Any green number

Any number that is red or green

Any number that is red and even

You are asked by a state government official to investigate whether under a proposed license plate system, there will be enough license plate codes for the state. There are 2 schemes proposed:

Each license plate is to show 3 letters, followed by 3 numbers, with no restrictions on repeated characters or their order. The only restriction is that the letters I and O are not to be used.

The second scheme is the same as above, except no letter or number can appear twice in the same license plate.

Under each scheme, how many different license plates can be produced?

Under which scheme is it more likely that there will be enough license plates? Explain.

Part I: Set Theory

Look up a roulette wheel diagram. The following sets are defined:

A = the set of red numbers

B = the set of black numbers

C = the set of green numbers

D = the set of even numbers

E = the set of odd numbers

F = {1,2,3,4,5,6,7,8,9,10,11,12}

From these, determine each of the following:

A∪B

A∩D

B∩C

C∪E

B∩F

E∩F

Part II: Relations and Functions

The implementation of the program that runs the game involves testing. One of the necessary tests is to see if the simulated spins are random. Create an n-ary relation, in table form, that depicts possible results of 10 trials of the game. Include the following results of the game:

Number

Color

Odd or even (note: 0 and 00 are considered neither even nor odd.)

Also include a primary key. What is the value of n in this n-ary relation?

Part III: Graphs and Trees

Create a tree that models the following scenario. A player decides to play a maximum of 4 times, betting on red each time. The player will quit after losing twice. In the tree, any possible last plays will be an ending point of the tree. Branches of the tree should indicate the winning or losing, and how that affects whether a new play is made.

Part IV: Combinatorics and Probability

In the roulette game, what is the probability of an outcome of:

Any odd number

Any green number

Any number that is red or green

Any number that is red and even

You are asked by a state government official to investigate whether under a proposed license plate system, there will be enough license plate codes for the state. There are 2 schemes proposed:

Each license plate is to show 3 letters, followed by 3 numbers, with no restrictions on repeated characters or their order. The only restriction is that the letters I and O are not to be used.

The second scheme is the same as above, except no letter or number can appear twice in the same license plate.

Under each scheme, how many different license plates can be produced?

Under which scheme is it more likely that there will be enough license plates? Explain.

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