To solve this problem, we need to calculate the probability of passing the quiz, which requires answering at least 60% of the questions correctly. The quiz consists of 20 multiple-choice questions, each with 4 possible answers, and the minimum passing grade is 60%.

Step 1: Determine the Number of Correct Answers Needed to Pass

The quiz has 20 questions, and passing requires answering at least 60% of the questions correctly: $\text{Passing grade} = 0.60 \times 20 = 12 \text{ correct answers}.$ So, to pass, a person must answer at least 12 questions correctly.

Step 2: Probability of Answering a Question Correctly

Since the person is guessing randomly, the probability of guessing a question correctly is: $P(\text{correct}) = \frac{1}{4} = 0.25.$ Thus, the probability of guessing incorrectly is: $P(\text{incorrect}) = 1 - P(\text{correct}) = 0.75.$

Step 3: Define the Situation as a Binomial Distribution

This is a binomial distribution problem because:

• There are 20 questions (trials),
• Each question has two possible outcomes (correct or incorrect),
• The probability of success (correct answer) is 0.25 for each trial.

The number of correct answers, $X$, follows a binomial distribution $X \sim \text{Binomial}(n = 20, p = 0.25)$.

We need to calculate the probability that the person gets at least 12 correct answers, i.e., $P(X \geq 12)$.

Step 4: Calculate the Probability of Passing

To find the probability of passing, we sum the probabilities of getting 12 or more correct answers: $P(X \geq 12) = P(X = 12) + P(X = 13) + \dots + P(X = 20).$ The probability mass function for a binomial distribution is given by: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}.$ Where:

• $\binom{n}{k}$ is the binomial coefficient,
• $p$ is the probability of a correct answer,
• $k$ is the number of correct answers.

This calculation involves multiple terms, so it's best done using a statistical calculator, a software tool like Python, or a binomial table.

Would you like me to calculate this for you or explain further how to use tools for this calculation?