To find the probability of getting 7 out of 10 questions correct when randomly guessing on true or false questions, we will use the binomial probability formula:
\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]
Where:
- \(n\) is the total number of trials (questions),
- \(x\) is the number of successful trials (correct answers),
- \(p\) is the probability of success on an individual trial,
- \(\binom{n}{x}\) is the binomial coefficient.
In this case:
- \(n = 10\) (total questions),
- \(x = 7\) (correct answers),
- \(p = 0.5\) (probability of guessing correctly for each question since there are two options: true or false).
Now, substitute the values into the formula:
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Calculate the binomial coefficient \(\binom{10}{7}\): \[ \binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
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Now calculate \(p^x (1-p)^{n-x}\): \[ p^7 = (0.5)^7 = \frac{1}{128} \] \[ (1-p)^{10-7} = (0.5)^3 = \frac{1}{8} \] Combine these: \[ p^7 (1-p)^{3} = \frac{1}{128} \times \frac{1}{8} = \frac{1}{1024} \]
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Finally, put it all together: \[ P(X = 7) = \binom{10}{7} p^7 (1-p)^{3} = 120 \times \frac{1}{1024} = \frac{120}{1024} = \frac{15}{128} \approx 0.1171875 \]
Thus, the probability that the student gets 7 questions right is approximately \(11.7%\).
The closest response option is 11.8%.