To find the probability that a student gets exactly 12 questions correct on a 20-question multiple choice test where each question has 5 answers, we can model this situation using the binomial probability formula.
In this scenario:
- The number of trials (questions) \( n = 20 \).
- The number of successes (correct answers) \( k = 12 \).
- The probability of success on a single trial (getting a question right) \( p = \frac{1}{5} \).
- The probability of failure (getting a question wrong) \( q = 1 - p = \frac{4}{5} \).
The binomial probability formula is given by:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
Where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).
Now substituting in our values:
- Calculate the binomial coefficient \( \binom{20}{12} \):
\[ \binom{20}{12} = \frac{20!}{12!(20-12)!} = \frac{20!}{12!8!} \]
Calculating \( \binom{20}{12} \):
This can also be expressed as \( \binom{20}{12} = \binom{20}{8} \) due to the property of binomial coefficients.
Using \( \binom{20}{8} \):
\[ \binom{20}{8} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 125970 \]
- Next, calculate \( p^{12} \) and \( q^{8} \):
\[ p^{12} = \left(\frac{1}{5}\right)^{12} = \frac{1}{5^{12}} = \frac{1}{244140625} \]
\[ q^{8} = \left(\frac{4}{5}\right)^{8} = \frac{4^8}{5^8} = \frac{65536}{390625} \]
Now, we can substitute these values into the binomial probability formula:
\[ P(X = 12) = \binom{20}{12} \cdot p^{12} \cdot q^{8} = 125970 \cdot \frac{1}{244140625} \cdot \frac{65536}{390625} \]
Calculating the product step-by-step:
\[ P(X = 12) = 125970 \cdot \frac{65536}{244140625 \cdot 390625} \]
Calculating \( 244140625 \times 390625 = 95367431640625 \):
Notice:
\[ P(X = 12) = 125970 \cdot \frac{65536}{95367431640625} \]
Now computing \( 125970 \cdot 65536 = 8251604480 \):
So finally:
\[ P(X = 12) = \frac{8251604480}{95367431640625} \]
This value can be calculated but is more conveniently left as a fraction. Evaluating this:
\[ P(X = 12) \approx 0.00008656 \]
Therefore, the final probability is:
\[ \boxed{0.0000866} \text{ or approximately 0.00866%} \]