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Sagot :
Alright, let's go through the problem step by step to determine the error in Jamal's prediction.
### Step 1: Understanding the Theoretical and Experimental Probability
1. Theoretical Probability:
- For a fair coin, the theoretical probability of landing heads up is [tex]\(0.5\)[/tex] (50%).
- Similarly, the theoretical probability of landing tails up is also [tex]\(0.5\)[/tex] (50%).
2. Experimental Probability:
- Jamal performed 10 trials, and the observed frequencies were:
- Heads: [tex]\(8\)[/tex] times
- Tails: [tex]\(2\)[/tex] times
- Therefore, the experimental probability from Jamal's experiment is:
[tex]\[ \text{Experimental Probability of Heads} = \frac{8}{10} = 0.80 \][/tex]
[tex]\[ \text{Experimental Probability of Tails} = \frac{2}{10} = 0.20 \][/tex]
### Step 2: Comparing Experimental Probability with Theoretical Probability
- The theoretical probability of heads is [tex]\(0.5\)[/tex].
- Jamal observed an experimental probability of heads [tex]\(0.80\)[/tex] from his 10 trials.
### Step 3: Error in Jamal’s Prediction
Jamal predicted that "the number of times the coin lands heads up will always be greater than the number of times it lands tails up" based on his limited trials. However, this prediction was made on experimental probability which can vary with a small number of trials. The theoretical probability, derived from the fairness of the coin, tells us that in the long run, the results should be closer to 50% heads and 50% tails.
Given the comparison:
- Experimental Probability of Heads (0.80) - Theoretical Probability of Heads (0.5) = Error in prediction.
- Error = [tex]\(0.80 - 0.50 = 0.30\)[/tex].
### Step 4: Determining the Adequacy of Trials
- With only 10 trials, the sample size is too small to accurately compare the experimental probability to the theoretical probability.
- A larger number of trials would yield results that better reflect the theoretical probability (closer to [tex]\(0.5\)[/tex] for both heads and tails).
### Conclusion
The error in Jamal's prediction is that he did not perform enough trials to make an accurate comparison between the theoretical and experimental probabilities. A larger number of trials would be necessary for his experimental probabilities to more closely match the theoretical probabilities.
### Final Choice
Therefore, the correct answer to the problem is:
- He did not perform enough trials to compare the theoretical and experimental probabilities.
### Step 1: Understanding the Theoretical and Experimental Probability
1. Theoretical Probability:
- For a fair coin, the theoretical probability of landing heads up is [tex]\(0.5\)[/tex] (50%).
- Similarly, the theoretical probability of landing tails up is also [tex]\(0.5\)[/tex] (50%).
2. Experimental Probability:
- Jamal performed 10 trials, and the observed frequencies were:
- Heads: [tex]\(8\)[/tex] times
- Tails: [tex]\(2\)[/tex] times
- Therefore, the experimental probability from Jamal's experiment is:
[tex]\[ \text{Experimental Probability of Heads} = \frac{8}{10} = 0.80 \][/tex]
[tex]\[ \text{Experimental Probability of Tails} = \frac{2}{10} = 0.20 \][/tex]
### Step 2: Comparing Experimental Probability with Theoretical Probability
- The theoretical probability of heads is [tex]\(0.5\)[/tex].
- Jamal observed an experimental probability of heads [tex]\(0.80\)[/tex] from his 10 trials.
### Step 3: Error in Jamal’s Prediction
Jamal predicted that "the number of times the coin lands heads up will always be greater than the number of times it lands tails up" based on his limited trials. However, this prediction was made on experimental probability which can vary with a small number of trials. The theoretical probability, derived from the fairness of the coin, tells us that in the long run, the results should be closer to 50% heads and 50% tails.
Given the comparison:
- Experimental Probability of Heads (0.80) - Theoretical Probability of Heads (0.5) = Error in prediction.
- Error = [tex]\(0.80 - 0.50 = 0.30\)[/tex].
### Step 4: Determining the Adequacy of Trials
- With only 10 trials, the sample size is too small to accurately compare the experimental probability to the theoretical probability.
- A larger number of trials would yield results that better reflect the theoretical probability (closer to [tex]\(0.5\)[/tex] for both heads and tails).
### Conclusion
The error in Jamal's prediction is that he did not perform enough trials to make an accurate comparison between the theoretical and experimental probabilities. A larger number of trials would be necessary for his experimental probabilities to more closely match the theoretical probabilities.
### Final Choice
Therefore, the correct answer to the problem is:
- He did not perform enough trials to compare the theoretical and experimental probabilities.
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