A quiz consists of 10 multiple-choice questions, each with 4 possible answers, only one of which is correct. A student who does not attend lectures on a regular basis has no clue what the answers are, and therefore uses an independent random guess to answer each of the 10 questions. What is the probability that the student gets at least one question right?Let's decide on some notations.Let L be the event that the student gets at least one of the questions right.We'll use R for getting a question right and W for getting it wrong (W is essentially "not R").Given the "tactic" of the student, what is P(R) and P(W)?
Question
A quiz consists of 10 multiple-choice questions, each with 4 possible answers, only one of which is correct. A student who does not attend lectures on a regular basis has no clue what the answers are, and therefore uses an independent random guess to answer each of the 10 questions. What is the probability that the student gets at least one question right?Let's decide on some notations.Let L be the event that the student gets at least one of the questions right.We'll use R for getting a question right and W for getting it wrong (W is essentially "not R").Given the "tactic" of the student, what is P(R) and P(W)?
Solution
Sure, let's break this down:
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First, we need to determine the probability of getting a question right (P(R)) and the probability of getting a question wrong (P(W)). Since there are 4 possible answers and only 1 is correct, P(R) = 1/4 = 0.25. Therefore, P(W) = 1 - P(R) = 0.75.
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Next, we need to find the probability of getting all questions wrong. Since the student is guessing independently for each question, we can simply raise P(W) to the power of the number of questions. So, the probability of getting all questions wrong is P(W)^10 = 0.75^10 ≈ 0.0563.
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However, we want to find the probability of getting at least one question right, which is the opposite of getting all questions wrong. Therefore, we subtract the probability of getting all questions wrong from 1. So, the probability of getting at least one question right is 1 - P(W)^10 = 1 - 0.0563 = 0.9437.
So, there is approximately a 94.37% chance that the student will get at least one question right by guessing randomly.
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