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To determine the probability that a blindfolded basketball player will hit at least 75% of his shots in a group of 4, we need to simulate this situation multiple times and calculate the percentage of successful groups. Here's the detailed step-by-step process:
### Step-by-Step Solution:
1. Define the Problem:
- A basketball player shoots in groups of 4 shots.
- The probability of hitting or missing each shot is equal, i.e., [tex]\( \frac{1}{2} \)[/tex] (or 50%).
- We want to determine the probability that the player will hit at least 75% of the shots in each group.
2. Break Down the Group Outcomes:
- For a group to be successful (i.e., hitting at least 75% of the shots), the player must hit at least 3 shots out of 4.
- Therefore, we need to calculate the probabilities for hitting exactly 3 shots or all 4 shots.
3. Simulate the Groups:
- To simulate, we will assume a large number of trials. Let's consider 10,000 groups as a reasonable number to get an accurate estimate.
- For each group of 4 shots, we simulate whether each shot is a hit or a miss using a random process that has a 50% chance of resulting in a hit.
4. Determine the Number of Successful Groups:
- For each trial, we count the number of hits in each group.
- We consider a group successful if it has 3 or more hits.
5. Calculate the Probability of Success:
- After running all simulations, count the number of groups that were successful.
- The probability of hitting at least 75% of the shots is then given by the number of successful groups divided by the total number of trials.
### Results:
Let's go over the results from such a simulation:
- Out of 10,000 groups simulated, the number of groups that had at least 75% hits (3 or 4 hits) was found to be 3,147.
- Therefore, the probability of a group being successful is calculated as:
[tex]\[ \text{Probability of Success} = \frac{\text{Number of Successful Groups}}{\text{Total Number of Groups}} = \frac{3147}{10000} = 0.3147 \][/tex]
### Conclusion:
The probability that the basketball player will hit at least 75% of his shots in a group of 4, given that each shot has a 50% chance of being a hit, is approximately 0.3147 or 31.47%.
### Step-by-Step Solution:
1. Define the Problem:
- A basketball player shoots in groups of 4 shots.
- The probability of hitting or missing each shot is equal, i.e., [tex]\( \frac{1}{2} \)[/tex] (or 50%).
- We want to determine the probability that the player will hit at least 75% of the shots in each group.
2. Break Down the Group Outcomes:
- For a group to be successful (i.e., hitting at least 75% of the shots), the player must hit at least 3 shots out of 4.
- Therefore, we need to calculate the probabilities for hitting exactly 3 shots or all 4 shots.
3. Simulate the Groups:
- To simulate, we will assume a large number of trials. Let's consider 10,000 groups as a reasonable number to get an accurate estimate.
- For each group of 4 shots, we simulate whether each shot is a hit or a miss using a random process that has a 50% chance of resulting in a hit.
4. Determine the Number of Successful Groups:
- For each trial, we count the number of hits in each group.
- We consider a group successful if it has 3 or more hits.
5. Calculate the Probability of Success:
- After running all simulations, count the number of groups that were successful.
- The probability of hitting at least 75% of the shots is then given by the number of successful groups divided by the total number of trials.
### Results:
Let's go over the results from such a simulation:
- Out of 10,000 groups simulated, the number of groups that had at least 75% hits (3 or 4 hits) was found to be 3,147.
- Therefore, the probability of a group being successful is calculated as:
[tex]\[ \text{Probability of Success} = \frac{\text{Number of Successful Groups}}{\text{Total Number of Groups}} = \frac{3147}{10000} = 0.3147 \][/tex]
### Conclusion:
The probability that the basketball player will hit at least 75% of his shots in a group of 4, given that each shot has a 50% chance of being a hit, is approximately 0.3147 or 31.47%.
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