Answer :
To determine which statement is true, let's carefully examine both theoretical and experimental probabilities of students participating in swimming.
### Theorical Probability
1. Theoretical Probability of Swimming:
- Each student flips a coin. Since flipping a fair coin results in heads (swimming) or tails (hiking) with an equal chance of 1/2 each, the theoretical probability for swimming is indeed:
[tex]\[ P(\text{swimming}) = \frac{1}{2} = 0.5 \][/tex]
### Experimental Probability
2. Experimental Probability of Swimming:
- We are given that 23 students chose swimming out of 50 students.
- The experimental probability is calculated as:
[tex]\[ P(\text{swimming})_{exp} = \frac{23}{50} = 0.46 \][/tex]
### Statement Comparison
Now, let's verify the statements:
1. Statement 1:
- "The theoretical probability of swimming, [tex]\( P(\text{swimming}) \)[/tex], is [tex]\( \frac{1}{2} \)[/tex], but the experimental probability is [tex]\( \frac{23}{27} \)[/tex]."
- This statement is not correct as the experimental probability is [tex]\( \frac{23}{50} \)[/tex], not [tex]\( \frac{23}{27} \)[/tex].
2. Statement 2:
- "The theoretical probability of swimming, [tex]\( P(\text{swimming}) \)[/tex], is [tex]\( \frac{23}{27} \)[/tex], but the experimental probability is [tex]\( \frac{1}{2} \)[/tex]."
- This statement is also incorrect because the theoretical probability should be [tex]\( \frac{1}{2} \)[/tex]. Additionally, the experimental probability is [tex]\( \frac{23}{50} \)[/tex], not [tex]\( \frac{1}{2} \)[/tex].
3. Statement 3:
- "The theoretical probability of swimming, [tex]\( P(\text{swimming}) \)[/tex], is [tex]\( \frac{23}{50} \)[/tex], but the experimental probability is [tex]\( \frac{1}{2} \)[/tex]."
- This statement contains an accurate theoretical probability of [tex]\( \frac{1}{2} \)[/tex]. However, it incorrectly states the experimental probability. Actually, the theoretical probability is [tex]\( \frac{1}{2} \)[/tex] and the experimental probability is [tex]\( \frac{23}{50} \)[/tex] or 0.46.
Upon reviewing all options, none of the statements seem entirely accurate. Thus, it would be better to conclude which hypothesis matches the original given data and calculations.
#### Correct Review (From Given Data and Calculations):
The theoretical probability of swimming is [tex]\( \frac{1}{2} \)[/tex], and the experimental probability is [tex]\( \frac{23}{50} \)[/tex], which means that none of the original statements hold true. However, the closest accurate relation that matches given theoretical and experimental data is:
Theoretical probability of swimming, [tex]\( P(\text{swimming}) \)[/tex], is [tex]\( \frac{1}{2} \)[/tex], but the experimental probability is [tex]\( \frac{23}{50} \)[/tex].
Thus, the appropriate interpretation of both probabilities while based on given selected options confirms:
Theoretical probability [tex]\( P(\text{swimming}) = 0.5 \)[/tex], and Experimental probability [tex]\( P(\text{swimming})_{exp} = 0.46 \)[/tex].
### Theorical Probability
1. Theoretical Probability of Swimming:
- Each student flips a coin. Since flipping a fair coin results in heads (swimming) or tails (hiking) with an equal chance of 1/2 each, the theoretical probability for swimming is indeed:
[tex]\[ P(\text{swimming}) = \frac{1}{2} = 0.5 \][/tex]
### Experimental Probability
2. Experimental Probability of Swimming:
- We are given that 23 students chose swimming out of 50 students.
- The experimental probability is calculated as:
[tex]\[ P(\text{swimming})_{exp} = \frac{23}{50} = 0.46 \][/tex]
### Statement Comparison
Now, let's verify the statements:
1. Statement 1:
- "The theoretical probability of swimming, [tex]\( P(\text{swimming}) \)[/tex], is [tex]\( \frac{1}{2} \)[/tex], but the experimental probability is [tex]\( \frac{23}{27} \)[/tex]."
- This statement is not correct as the experimental probability is [tex]\( \frac{23}{50} \)[/tex], not [tex]\( \frac{23}{27} \)[/tex].
2. Statement 2:
- "The theoretical probability of swimming, [tex]\( P(\text{swimming}) \)[/tex], is [tex]\( \frac{23}{27} \)[/tex], but the experimental probability is [tex]\( \frac{1}{2} \)[/tex]."
- This statement is also incorrect because the theoretical probability should be [tex]\( \frac{1}{2} \)[/tex]. Additionally, the experimental probability is [tex]\( \frac{23}{50} \)[/tex], not [tex]\( \frac{1}{2} \)[/tex].
3. Statement 3:
- "The theoretical probability of swimming, [tex]\( P(\text{swimming}) \)[/tex], is [tex]\( \frac{23}{50} \)[/tex], but the experimental probability is [tex]\( \frac{1}{2} \)[/tex]."
- This statement contains an accurate theoretical probability of [tex]\( \frac{1}{2} \)[/tex]. However, it incorrectly states the experimental probability. Actually, the theoretical probability is [tex]\( \frac{1}{2} \)[/tex] and the experimental probability is [tex]\( \frac{23}{50} \)[/tex] or 0.46.
Upon reviewing all options, none of the statements seem entirely accurate. Thus, it would be better to conclude which hypothesis matches the original given data and calculations.
#### Correct Review (From Given Data and Calculations):
The theoretical probability of swimming is [tex]\( \frac{1}{2} \)[/tex], and the experimental probability is [tex]\( \frac{23}{50} \)[/tex], which means that none of the original statements hold true. However, the closest accurate relation that matches given theoretical and experimental data is:
Theoretical probability of swimming, [tex]\( P(\text{swimming}) \)[/tex], is [tex]\( \frac{1}{2} \)[/tex], but the experimental probability is [tex]\( \frac{23}{50} \)[/tex].
Thus, the appropriate interpretation of both probabilities while based on given selected options confirms:
Theoretical probability [tex]\( P(\text{swimming}) = 0.5 \)[/tex], and Experimental probability [tex]\( P(\text{swimming})_{exp} = 0.46 \)[/tex].
