AP2.25 - The Practice of Statistics for AP

Question

AP2.25. Five cards, each with a different symbol, are shuffled and you choose one. If it is the diamond, you win $\backslash (5.00$. The cards are reshuffled after each draw. You must pay $\backslash )1.00$ for each selection. You continue to play until you select the diamond. Is this a fair game (that is, on average
will you win the same amount as you lose)?

 (a) Describe how you will carry out a simulation of this game using the random digit table below. Be sure to indicate what the digits will represent. 

(b) Perform $10$ repetitions of your simulation. Copy the random digit table onto your paper. Mark on or above the table so that someone can follow your work.

12975   13258   13048   45144   72321   81940   00360 

02428  96767  35964   23822  96012  94591   65194 

50842  53372  72829   50232  97892  63408 77919 

44575  24870   04178   81565  42628  17797   49376 

61762   16953    88604  12724   62964  88145  83083  

69453  46109   59505  69680 00900  19687  12633

57857   95806   09931  02150  43163

(c) Based on your simulation, what is the average number of cards you would need to draw in order to obtain a diamond? Justify your answer.
(d) Is this a fair game (that is, on average will you win the same amount as you lose)? Explain your reasoning.

Step-by-Step Solution

Answer

(a) Since, $5$is the maximum number, only one digit is required. If the number is $0-4$, the cards will be selected; otherwise, it will continue on to the next random number.

(b) Performed $10$ repetitions of simulation as:

Repetition 1: $1, 2,1,3,2,1,3,0,4$ (Stop at $4$). Length: $9$. Profit $=5-9=-4$

 Repetition 2: (continue where left off) $4$ (stop). Length $=1$, Payout $=5-2=4$

Repetition 3: $1, 4$(stop). Length $=2$, payout $=5-3=3$

Repetition 4: $4$ (stop). Length $=1$, payout$=5-1=4$

Repetition 5:$2,3,2,1,1,4$ (stop). Length. Payout$=5-6=-1$

Repetition 6:  (stop), length $=8$, payout $=5-8=-3$

Repetition 7: $2,3, 4$(stop). Length $=3$. Payout$=5-3=2$

Repetition 8: $2,3,2,2,0,1,2,4$ (stop). Length $=8$. Payout $=5-8=-3$

Repetition 9: $2,3,2,2,0,1,2,4$ (stop). Length $=8$. Payout $=5-8=-3$

Repetition 10: $1,1,4$ (stop). Length $=3$. Payout $=5-3=2$

(c) The average number of simulation is $4.9$ draws.

(d) The simulation supports this, and it would be a fair game.

Part (a) Step 1: Given information

To describe how the random digit table will be used to simulate this game. And then to indicate what the digits will represent.

Part (a) Step 2: Explanation

Since the five cards are shuffled, each with a different symbol, and you must choose one. Will win if it's a diamond.
As a result, a simulation of this game will be run using a random digit table, as the labels will be assigned according on the order in which they appeared in the image, but this will be arbitrary.
So, for each form, you'll need numeric labels first.
Therefore,
Star$=0$, Hexagon $=1$, Circle $=2$, Triangle $=3$, Diamond $=4$
Because $5$ is the maximum number, only one digit is required.
If the number is $0-4$, the cards will be selected; otherwise, it will continue on to the next random number.

Part (b) Step 1: Given information

To perform $10$ repetitions of the simulation. And to copy the random digit table onto the paper.

Part(b) Step 2: Explanation

The question asks to shuffle five cards, each with a different symbol, and choose one. Diamond will be victorious.
As a result, the payout each repetition will equal the number of random numbers minus from the number of draws.
If it takes three draws, for example, $5-3=2$.
If ten draws were required, $5-10=-5$.
This is because you must pay $5$ for each selection and receive $5$ at the end of the game.
So, after each repetition, a card will be picked, followed by the next random number until a diamond is drawn.
The payments for each iteration will be calculated.
And, as previously stated, the digits $5-9$ are not included (a).

Part (b) Step 3: Explanation

Asa result, 

Repetition 1: $1,2,1,3,2,1,3,0,4$ (Stop at $4$). Length: $9$ . Profit $=5-9=-4$
Repetition 2: (continue where left off) $4$ (stop). Length $=1$, Payout $=5-2=4$
Repetition 3: $1,4$ (stop). Length $=2$, payout $=5-3=3$
Repetition 4: $4$ (stop). Length $=1$, payout $=5-1=4$
Repetition 5: $2,3,2,1,1,4$(stop). Length $=6$. Payout $=5-6=-1$
Repetition 6: $0,0,0,3,0,0,2,4$ (stop), length$=8$, payout $=5-8=-3$
Repetition 7: $2,3,4$ (stop). Length $=3$. Payout $=5-3=2$
Repetition 8: $2,3,2,2,0,1,2,4$ (stop). Length $=8$. Payout $=5-8=-3$
Repetition 9: $2,3,2,2,0,1,2,4$ (stop). Length $=8$. Payout $=5-8=-3$
Repetition 10: $1,1,4$ (stop). Length $=3$. Payout $=5-3=2$.

Part (c) Step 1: Given information

To justify the answer that based on the simulation, and find the average number of cards would need to draw in order to obtain a diamond.

Part (c) Step 2: Explanation

The question asks to shuffle five cards, each with a different symbol, and choose one. If it's a diamond, you will win.
As a result, named the number of cards in each simulation length.
To calculate the average, put all ten repetitions together and divide by $10$. And the total length of the previous section is $49$.
So, having,
Average length$=\frac{49}{10}$

$=4.9$

As a result, the average number of simulation is $4.9$ draws.

Part (d) Step 1: Given information

To explain with reason that this is a fair game, that is, on average will  win the same amount as lose.

Part (d) Step 2: Explanation

The question states that five cards are shuffled, each with a different symbol, and we must choose one. We win if it's a diamond. 

As a result, we will use the simulation findings to address this question. We'd like to increase the repeat sizes in order to achieve a more precise answer. While the data is sampled at random, the sample size is small enough that sampling error could be significant. 

And the decision rule would be that we would break even if we drew an average of five cards per game. We drew roughly $4.9$ cards on average based on our simulation results. We'd say this is close to $5$ because we drew $4.9$ cards on average.

We'd have a $1$ in $5$ chance of acquiring the diamond if the numbers were genuinely random, and the expected loss would be zero. The simulation supports this, and it would be a fair game.