Judy’s situation is quite common for students who find themselves unprepared for a test. Let’s explore how we can calculate the probability of her guessing correctly.
Each question on the test presents 4 possible answers, from which only one is correct. Therefore, for each question, Judy has a 1 in 4 chance of selecting the correct answer. This probability can be expressed as:
Probability of guessing a single question correctly = 1/4
Now, since the test consists of 10 questions, and assuming each question is independent of the others, we can use the probability of guessing correctly for all questions.
For 10 independent questions, the formula to find the probability of guessing correctly on all questions is:
P(All Correct) = (1/4) ^ 10
This means Judy has a 1 in 1,048,576 chance of correctly answering all questions by guessing. In decimal form, this probability is:
P(All Correct) ≈ 0.0000009537
Now, if we also consider how many questions Judy might answer correctly, we utilize the binomial probability formula:
P(X = k) = (n choose k) * (p^k) * ((1 – p)^(n – k))
Where:
- n = total number of questions (10)
- k = number of correct answers (varies from 0 to 10)
- p = probability of guessing one question correctly (1/4)
Using this formula, she can find the probability of answering a specific number of questions correctly — be it 0, 1, 2, … up to 10. Nevertheless, the overall long shot is that Judy could luck out and ace the test by randomly guessing, though the odds are considerably against her. In essence, while Judy may have a chance of getting some answers correct by sheer luck, her overall likelihood of scoring well by guessing all answers is incredibly low.