Question

A student randomly guesses on 10 true or false questions. Use the binomial model to determine the probability that the student gets 5 out of the 10 questions right. Show all your steps. P(x)=[n!x!(n−x)!]pxqn−x (4 points)
short answer

Answer 1

To determine the probability that a student randomly guesses and gets exactly 5 out of 10 true or false questions correct, we can use the binomial probability formula:

\[
P(x) = \binom{n}{x} p^x q^{n-x}
\]

where:
- \( n \) is the total number of questions (in this case, \( n = 10 \))
- \( x \) is the number of correct answers (in this case, \( x = 5 \))
- \( p \) is the probability of getting a question right (for true/false questions, \( p = 0.5 \))
- \( q \) is the probability of getting a question wrong (since \( q = 1 - p\), we have \( q = 0.5 \))

### Step 1: Calculate \( \binom{n}{x} \)

The binomial coefficient \( \binom{n}{x} \) (which is read as "n choose x") is calculated as follows:

\[
\binom{n}{x} = \frac{n!}{x!(n-x)!}
\]

Substituting the values \( n = 10 \) and \( x = 5 \):

\[
\binom{10}{5} = \frac{10!}{5! \cdot (10-5)!} = \frac{10!}{5! \cdot 5!}
\]

Calculating the factorials:

- \( 10! = 3628800 \)
- \( 5! = 120 \)

So,

\[
\binom{10}{5} = \frac{3628800}{120 \cdot 120} = \frac{3628800}{14400} = 252
\]

### Step 2: Calculate \( p^x \) and \( q^{n-x} \)

Now calculate \( p^x \) and \( q^{n-x} \):

\[
p^x = (0.5)^5 = \frac{1}{32}
\]

\[
q^{n-x} = (0.5)^{10-5} = (0.5)^5 = \frac{1}{32}
\]

### Step 3: Combine everything into the formula

Now substitute everything into the binomial probability formula:

\[
P(5) = \binom{10}{5} \cdot p^5 \cdot q^{5} = 252 \cdot \left(\frac{1}{32}\right) \cdot \left(\frac{1}{32}\right)
\]

Now calculate the total probability:

\[
P(5) = 252 \cdot \left(\frac{1}{32}\right)^2 = 252 \cdot \frac{1}{1024} = \frac{252}{1024}
\]

Simplifying \( \frac{252}{1024} \):

Both numbers are divisible by 4:

\[
\frac{252 \div 4}{1024 \div 4} = \frac{63}{256}
\]

Thus, the probability that the student gets exactly 5 questions correct out of 10 is:

\[
\boxed{\frac{63}{256}}
\]