Consider an experiment in which 12 identical small boxes are placed side by side on a table. A crystal is placed at random inside one of the boxes. A self-professed "psychic" is asked to pick the box that contains the crystal. If the experiment is repeated 9 times, what is the probability that psychic guesses correctly at least once?

To solve this problem, we need to calculate the probability that the psychic guesses correctly **at least once** out of 9 trials, where in each trial, the psychic selects one box from 12 identical boxes.

### Step 1: Probability of a Correct Guess in One Trial
In each trial, since there are 12 identical boxes and the crystal is randomly placed in one of them, the probability of the psychic guessing correctly in a single trial is:

\[
P(\text{correct guess in one trial}) = \frac{1}{12}
\]

### Step 2: Probability of a Wrong Guess in One Trial
The probability that the psychic **does not** guess correctly in one trial is:

\[
P(\text{wrong guess in one trial}) = 1 - \frac{1}{12} = \frac{11}{12}
\]

### Step 3: Probability of Wrong Guess in All 9 Trials
Now, we need to find the probability that the psychic guesses incorrectly in **all 9 trials**. Since each trial is independent, the probability of making a wrong guess in all 9 trials is:

\[
P(\text{wrong guess in all 9 trials}) = \left( \frac{11}{12} \right)^9
\]

### Step 4: Probability of at Least One Correct Guess
To find the probability that the psychic guesses correctly **at least once**, we can subtract the probability of guessing incorrectly in all 9 trials from 1:

\[
P(\text{correct at least once}) = 1 - P(\text{wrong guess in all 9 trials}) = 1 - \left( \frac{11}{12} \right)^9
\]

### Step 5: Calculating the Result
Now, we compute the value of \(\left( \frac{11}{12} \right)^9\):

\[
\left( \frac{11}{12} \right)^9 \approx 0.42257
\]

Thus, the probability of guessing correctly at least once is:

\[
P(\text{correct at least once}) = 1 - 0.42257 = 0.57743
\]

### Final Answer:
The probability that the psychic guesses correctly at least once in 9 trials is approximately:

\[
P(\text{correct at least once}) \approx 0.577
\]

Would you like more details or clarifications on any part of the solution?

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Here are 5 related questions to deepen your understanding:
1. How would the probability change if there were 10 boxes instead of 12?
2. What if the experiment was repeated 5 times instead of 9?
3. How does the probability change as the number of boxes increases indefinitely?
4. What if the psychic had to guess correctly twice instead of at least once?
5. Can we generalize this to a situation with \(n\) boxes and \(k\) trials?

**Tip:** The probability of success increases significantly as the number of trials grows, even when individual trial probabilities are small.

Consider an experiment in which 12 identical small boxes are placed side by side on a table. A crystal is placed at random inside one of the boxes. A self-professed 'psychic' is asked to pick the box that contains the crystal. If the experiment is repeated 9 times, what is the probability that the psychic guesses correctly at least once?

Discover how to calculate the probability that a psychic guesses correctly at least once in 9 trials, with a crystal placed randomly in one of 12 identical boxes. The solution explores key probability concepts, including independent events and the complement rule. Suitable for high school students.

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