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)

Answer 1

To determine the probability that a student gets exactly 5 out of 10 true or false questions correct by guessing, we can use the binomial probability formula. The binomial formula is given by:

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

where:
- \( n \) is the number of trials (questions),
- \( x \) is the number of successes (correct answers),
- \( p \) is the probability of success on each trial,
- \( q \) is the probability of failure on each trial,
- \( \binom{n}{x} \) is the binomial coefficient, calculated as \(\binom{n}{x} = \frac{n!}{x!(n-x)!}\).

### Given:
- \( n = 10 \) (total questions),
- \( x = 5 \) (questions answered correctly),
- Since it’s a true or false question, \( p = 0.5 \) (probability of guessing correctly),
- \( q = 1 - p = 0.5 \) (probability of guessing incorrectly).

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

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

Calculating the factorials:
- \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5! \)
- Thus, \( \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5! \times 5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{120} = 252 \)

### Step 2: Substitute the values into the binomial formula

Now we can compute \( P(X = 5) \):

\[
P(X = 5) = \binom{10}{5} (0.5)^5 (0.5)^{10-5}
\]

### Step 3: Simplify the expression

\[
P(X = 5) = 252 \cdot (0.5)^5 \cdot (0.5)^5 = 252 \cdot (0.5)^{10}
\]
\[
= 252 \cdot \frac{1}{1024}
\]
\[
= \frac{252}{1024}
\]

### Step 4: Reduce the fraction

To simplify \(\frac{252}{1024}\), we find the greatest common divisor (GCD) for 252 and 1024.

By dividing both the numerator and the denominator by 4:

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

### Conclusion

The final answer is:

\[
P(X = 5) = \frac{63}{256} \approx 0.2461
\]

Thus, the probability that the student gets exactly 5 out of 10 true or false questions correct while guessing is approximately \(0.2461\) or \(24.61\%\).