Statistical Models: Mastery Test

Rick organized three games for a club's [tex]$25^{\text {th}}$[/tex] anniversary celebration.

A. Get-it-Rolling is a game in which the player rolls an eight-sided die. The player wins if the die lands on 5.
B. Bag-of-Tokens is a game in which the player draws a token from a bag containing 7 tokens, each of a different color. The player wins if they draw the red token.
C. Pick-Your-Tile is a game in which the player picks a number tile from a box of 12 tiles, each with a different number on it. The player wins if they pick the tile that has the number 9 on it.

Rick kept track of wins and losses for each game attempt in the following table.

\begin{tabular}{|c|c|c|}
\hline Game & Number of Wins & Number of Losses \\
\hline Get-it-Rolling (A) & 26 & 173 \\
\hline Bag-of-Tokens (B) & 54 & 141 \\
\hline Pick-Your-Tile (C) & 17 & 175 \\
\hline
\end{tabular}

Select the correct statement.

A. Only the results from game A align closely with the theoretical probability of winning that game.
B. The results from both game A and game C align closely with the theoretical probability of winning those games, while the results from game B do not.
C. Only the results from game B align closely with the theoretical probability of winning that game.
D. The results from both game B and game C align closely with the theoretical probability of winning those games, while the results from game A do not.



Answer :

Let's analyze the games and their probabilities in detail.

### Game Details and Theoretical Probabilities:

Game A: Get-it-Rolling
- Win condition: Rolling a 5 on an eight-sided die.
- Theoretical probability of winning (P_A_theoretical): [tex]\( \frac{1}{8} \)[/tex].

Game B: Bag-of-Tokens
- Win condition: Drawing the red token from a bag of 7 different colored tokens.
- Theoretical probability of winning (P_B_theoretical): [tex]\( \frac{1}{7} \)[/tex].

Game C: Pick-Your-Tile
- Win condition: Picking the tile with the number 9 out of 12 tiles.
- Theoretical probability of winning (P_C_theoretical): [tex]\( \frac{1}{12} \)[/tex].

### Experimental Probabilities:

From the given data table:

1. Get-it-Rolling (A)
- Wins: 26
- Losses: 173
- Total attempts: [tex]\( 26 + 173 = 199 \)[/tex]
- Experimental probability (P_A_experimental): [tex]\( \frac{26}{199} \)[/tex].

2. Bag-of-Tokens (B)
- Wins: 54
- Losses: 141
- Total attempts: [tex]\( 54 + 141 = 195 \)[/tex]
- Experimental probability (P_B_experimental): [tex]\( \frac{54}{195} \)[/tex].

3. Pick-Your-Tile (C)
- Wins: 17
- Losses: 175
- Total attempts: [tex]\( 17 + 175 = 192 \)[/tex]
- Experimental probability (P_C_experimental): [tex]\( \frac{17}{192} \)[/tex].

### Deviation Calculation:

To determine how closely each game's experimental probability aligns with its theoretical probability, we calculate the absolute deviations:

1. Get-it-Rolling (A):
- Deviation: [tex]\( \left| \frac{1}{8} - \frac{26}{199} \right| \)[/tex].

2. Bag-of-Tokens (B):
- Deviation: [tex]\( \left| \frac{1}{7} - \frac{54}{195} \right| \)[/tex].

3. Pick-Your-Tile (C):
- Deviation: [tex]\( \left| \frac{1}{12} - \frac{17}{192} \right| \)[/tex].

We then compare these deviations against a small tolerance level ([tex]\(\epsilon = 0.05\)[/tex]), which represents a close alignment.

### Conclusion:

Based on our analysis, here's how the games align with their theoretical probabilities:
- Game A's results align closely.
- Game B's results do not align closely.
- Game C's results align closely.

### Decision:

Given these observations:

Option B is correct:
"The results from both game A and game C align closely with the theoretical probability of winning those games, while the results from game B do not."

Thus, the correct answer is:
B. The results from both game A and game [tex]$C$[/tex] align closely with the theoretical probability of winning those games, while the results from game B do not.