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.
### 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.