Statistics for my University Coursework

[Drrgh]

This is one of the opening questions that happen in my assignment, and if someone could quickly walk me through how or what I am supposed to do I would be greatly in your debt.

Question 2
John has two wheels of fortune. The first wheel contains 12 equally-sized parts; three of these parts contain the number 0, four contain a 3, and five contain a 5. The second wheel contains 18 equally-sized parts, of which five parts contain a 0, three contain a 1, three contain a 2, three contain a 4, two contain a 6, and two contain an 8. Both wheels are fair. John is at a flea market and uses these wheels as follows: the first wheel determines the price the player has to pay, and the second wheel determines the amount the player wins.

2a (14 points)
Give the probability distribution functions of both wheels of fortune. What are the expected payoffs and variances of both wheels? How much can a player expect to win by playing the game once, and what is the variance of this expected profit?

So far I have tried charting the numbers with the sections per wheel and trying to find variance, but each time I do this I find different answers using my graphical calculator, excel and manually going through it.

Here is a spreadsheet I was trying to use, thanks for any direction in advance.

 

[pka]

This is one of the opening questions that happen in my assignment, and if someone could quickly walk me through how or what I am supposed to do I would be greatly in your debt.
Question 2
John has two wheels of fortune. The first wheel contains 12 equally-sized parts; three of these parts contain the number 0, four contain a 3, and five contain a 5. The second wheel contains 18 equally-sized parts, of which five parts contain a 0, three contain a 1, three contain a 2, three contain a 4, two contain a 6, and two contain an 8. Both wheels are fair. John is at a flea market and uses these wheels as follows: the first wheel determines the price the player has to pay, and the second wheel determines the amount the player wins.
2a (14 points)
Give the probability distribution functions of both wheels of fortune. What are the expected payoffs and variances of both wheels? How much can a player expect to win by playing the game once, and what is the variance of this expected profit?

No spreadsheets here. I will help with the setup.
Let \(\displaystyle X\) be the value on the first wheel and \(\displaystyle Y\) the value on the second.
\(\displaystyle \mathcal{P}(X=0)=\frac{3}{12}\). WHY?

\(\displaystyle (X-Y)\) is equal to John's gain or loss.
I think in terms of ordered pairs \(\displaystyle (X,Y)\) of which there are eighteen.
BUT there are not eighteen values for \(\displaystyle X-Y\) WHY?

\(\displaystyle {\large\mathcal{P}(X-Y=1) =\mathcal{P}(X=3)\mathcal{P}(Y=2)+\mathcal{P}(X=5) \mathcal{P}(Y=4)}\)

 

[Ishuda]

This is one of the opening questions that happen in my assignment, and if someone could quickly walk me through how or what I am supposed to do I would be greatly in your debt.

Question 2
John has two wheels of fortune. The first wheel contains 12 equally-sized parts; three of these parts contain the number 0, four contain a 3, and five contain a 5. The second wheel contains 18 equally-sized parts, of which five parts contain a 0, three contain a 1, three contain a 2, three contain a 4, two contain a 6, and two contain an 8. Both wheels are fair. John is at a flea market and uses these wheels as follows: the first wheel determines the price the player has to pay, and the second wheel determines the amount the player wins.

2a (14 points)
Give the probability distribution functions of both wheels of fortune. What are the expected payoffs and variances of both wheels? How much can a player expect to win by playing the game once, and what is the variance of this expected profit?

So far I have tried charting the numbers with the sections per wheel and trying to find variance, but each time I do this I find different answers using my graphical calculator, excel and manually going through it.

Here is a spreadsheet I was trying to use, thanks for any direction in advance.

I assume that the 0, 3, or 5 is the amount you would pay to bet and the 0, 1, 2, 4, 6, 8 is how much you would win. So spin the first wheel get any of the 0 (dollars, pounds, euros, ....) slots you would pay zero. Then if you got any of 8 slots you would receive 8.

There are 12*18 outcomes. That is, if (b_i, w_i) are pairs of bets and wins then there are 216 of them (or, not counting duplicates, 18 of them). For example the probability of (0, 8) [having to pay zero but win eight] would be
P₁(0) * P₂(8) = (3/12) * (2/18) = 6 / 216.

So, for your spread sheet, you would lay the first wheel in a row and the second wheel in a column [for display purposes]:

			FIRST WHEEL
SECOND WHEEL	Bet	0	3	5
Win	Probability	Probability	0.25	0.33	0.42
0	0.28
1	0.17
2	0.17
…

Now fill in the table. The probability that you have to bet 3 would be the sum of the FIRST WHEEL 3 column. The probability that you would win at least 4 would be the sum of the 4, 6, and 8 rows of the SECOND WHEEL.