Question
A student randomly guesses on 10 true or false questions. Use the binomial model to find the probability that the student gets 7 out of the 10 questions right. P(x)=[n!x!(n−x)!]pxqn−x (1 point) Responses 11.8% 11.8% 20.9% 20.9% 32.4% 32.4% 50%
Answers
GPT-4o mini
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:
1. Calculate the binomial coefficient \(\binom{10}{7}\):
\[
\binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]
2. 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:
\[
p^7 (1-p)^{3} = \frac{1}{128} \times \frac{1}{8} = \frac{1}{1024}
\]
3. 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%**.
\[
P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}
\]
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:
1. Calculate the binomial coefficient \(\binom{10}{7}\):
\[
\binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
\]
2. 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:
\[
p^7 (1-p)^{3} = \frac{1}{128} \times \frac{1}{8} = \frac{1}{1024}
\]
3. 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%**.