Answer :
Let's examine Elijah's prediction step by step to understand whether it is reasonable or not based on theoretical probability.
1. Number of Spins:
Elijah plans to spin the spinner 180 times.
2. Elijah's Prediction:
Elijah predicts that the spinner will land on the section labeled 3 exactly 60 times out of the 180 spins.
3. Theoretical Probability:
The theoretical probability of the spinner landing on any specific section can be calculated assuming the spinner is fair and has evenly divided sections. If there are 3 sections on the spinner, each section would have an equal probability of [tex]\(\frac{1}{3}\)[/tex] (or approximately 0.3333).
4. Expected Number of Spins Landing on 3:
Using the theoretical probability, we can calculate the expected number of spins that result in landing on the section labeled 3. This is done by multiplying the total number of spins by the theoretical probability:
[tex]\[ \text{Expected spins landing on 3} = 180 \times \frac{1}{3} = 60 \][/tex]
5. Comparison:
Compare Elijah’s predicted number of spins (60) with the expected number of spins based on the theoretical probability (60). We observe that they are equal:
[tex]\[ 60 \text{ (predicted)} = 60 \text{ (expected)} \][/tex]
6. Conclusion:
Since Elijah's prediction of 60 spins landing on the section labeled 3 matches exactly with the expected number of spins predicted by the theoretical probability, we can conclude that Elijah's prediction is reasonable and accurate.
Therefore, Elijah's prediction is exact because it matches perfectly with what we would theoretically expect based on the probability distribution of the spinner.
1. Number of Spins:
Elijah plans to spin the spinner 180 times.
2. Elijah's Prediction:
Elijah predicts that the spinner will land on the section labeled 3 exactly 60 times out of the 180 spins.
3. Theoretical Probability:
The theoretical probability of the spinner landing on any specific section can be calculated assuming the spinner is fair and has evenly divided sections. If there are 3 sections on the spinner, each section would have an equal probability of [tex]\(\frac{1}{3}\)[/tex] (or approximately 0.3333).
4. Expected Number of Spins Landing on 3:
Using the theoretical probability, we can calculate the expected number of spins that result in landing on the section labeled 3. This is done by multiplying the total number of spins by the theoretical probability:
[tex]\[ \text{Expected spins landing on 3} = 180 \times \frac{1}{3} = 60 \][/tex]
5. Comparison:
Compare Elijah’s predicted number of spins (60) with the expected number of spins based on the theoretical probability (60). We observe that they are equal:
[tex]\[ 60 \text{ (predicted)} = 60 \text{ (expected)} \][/tex]
6. Conclusion:
Since Elijah's prediction of 60 spins landing on the section labeled 3 matches exactly with the expected number of spins predicted by the theoretical probability, we can conclude that Elijah's prediction is reasonable and accurate.
Therefore, Elijah's prediction is exact because it matches perfectly with what we would theoretically expect based on the probability distribution of the spinner.
