Find the probability of guessing correctly at least nine out of ten answers in a "true" or "false" objective test.

Answer

To find the probability of guessing correctly at least nine out of ten answers in a "true" or "false" objective test, we can follow these steps:

### Step 1: Understand the Problem
We need to calculate the probability of getting at least 9 correct answers out of 10 questions, where each question has two possible answers: True (T) or False (F).

### Step 2: Total Possible Outcomes
For each question, there are 2 possible answers. Therefore, for 10 questions, the total number of ways to answer them is:
\[ 2^{10} \]
Calculating this gives:
\[ 2^{10} = 1024 \]

### Step 3: Calculate Favorable Outcomes
We need to find the number of ways to guess correctly at least 9 answers. This includes two scenarios: guessing exactly 9 correct answers and guessing all 10 correct answers.

#### Scenario 1: Guessing Exactly 9 Correct Answers
To find the number of ways to get exactly 9 correct answers, we can choose 9 questions to be correct from the 10 questions. The remaining 1 question will be incorrect. The number of ways to choose 9 questions from 10 is given by the binomial coefficient:
\[ \binom{10}{9} = 10 \]
This means there are 10 ways to guess exactly 9 correct answers.

#### Scenario 2: Guessing All 10 Correct Answers
There is only 1 way to get all 10 answers correct:
\[ \binom{10}{10} = 1 \]

### Step 4: Total Favorable Outcomes
Now, we add the number of favorable outcomes from both scenarios:
\[ \text{Total favorable outcomes} = \binom{10}{9} + \binom{10}{10} = 10 + 1 = 11 \]

### Step 5: Calculate the Probability
The probability of guessing at least 9 correct answers is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{at least 9 correct}) = \frac{\text{Total favorable outcomes}}{\text{Total possible outcomes}} = \frac{11}{1024} \]

### Final Answer
Thus, the probability of guessing correctly at least 9 out of 10 answers is:
\[ \frac{11}{1024} \]

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