Final answer:
The probability that an unprepared student gets exactly 4 questions correct on a 20 question multiple-choice exam with 5 answer choices is approximately 21.8%.
Step-by-step explanation:
To calculate the probability that an unprepared student gets exactly 4 questions correct on a 20 question multiple-choice exam with 5 answer choices each, we use the binomial probability formula:
P(X = k) = C(n, k) * (p)^k * (1-p)^(n-k)
Where:
- n is the number of trials (in this case, 20 questions),
- k is the number of successes (in this case, 4 correct answers),
- p is the probability of success on a single trial (with 5 choices, this is 1/5 or 0.2), and
- C(n, k) is the combination of n things taken k at a time.
Applying these to our situation:
P(X = 4) = C(20, 4) * (0.2)^4 * (0.8)^(16)
The combination C(20, 4) can be calculated as:
C(20, 4) = 20! / (4! * (20 - 4)!
Which simplifies to:
C(20, 4) = 4845
Therefore, the probability is:
P(X = 4) = 4845 * (0.2)^4 * (0.8)^(16)
Performing the calculations:
P(X = 4) ≈ 0.218199
So, the probability that the student gets exactly 4 questions correct is approximately 0.218 or 21.8%.