Question

A person answers each of two multiple choice questions at random. If there are four possible choices on each question, what is the conditional probability that both answers are correct given that at least one is correct?

Short answer

The conditional probability that both answers are correct given that at least one is correct, is \(P(A | B) \approx 0.143\).

Step-by-step solution

Identify and calculate probabilities

Define A as the event that both answers are correct, and B as the event that at least one answer is correct. Let's first figure out the probabilities for these two events. Since there are 4 choices and only 1 is correct, the probability of getting the answer correct (for a single question) is \(1/4 = 0.25\). Since the questions are independent, the probability of getting both answers correct (event A) is \((1/4) * (1/4) = 0.0625\).

Determine the probability of at least one correct answer

To find the probability of event B, we can first calculate the probability of its complement (both answers being wrong). The probability of getting a question wrong is \((1 - 1/4) = 3/4\), so the chance of getting both questions wrong is \((3/4) * (3/4) = 0.5625\). Thus, the probability of at least one answer being correct is \(1 - 0.5625 = 0.4375\).

Apply the conditional probability formula

Now, we can apply the conditional probability formula: \(P(A | B) = P(A \cap B) / P(B)\). In this scenario, \(P(A \cap B)\) is simply the probability of A, because A (both answers being correct) is a subset of B (at least one answer is correct). Hence, \(P(A | B) = 0.0625 / 0.4375\).

Key concepts

Multiple Choice Questions

Multiple choice questions are a common format in exams and quizzes where a question is followed by several possible answers, and the students must select the correct one among them. To address multiple choice questions effectively, understanding the structure of the question and analyzing each option is crucial. When guessing the answer to a multiple choice question, such as in our exercise, the probability of choosing the correct answer is determined by the number of choices presented.

In the given exercise, each question presents four possible answers, but only one is correct. Therefore, the probability of randomly guessing the correct answer for one question is \(\frac{1}{4} = 0.25\). The questions in this scenario are considered to be independent events, meaning the outcome of one question does not affect the outcome of the other.

Independent Events

In probability theory, independent events are events where the occurrence of one event does not influence the outcome of another. This is an important concept to understand when dealing with multiple random events, like answering multiple choice questions. When events are independent, the probability of both or all events happening is the product of their individual probabilities.

Using the exercise as an example, because each question is answered without any impact on the other, getting both questions correct involves multiplying their independent probabilities: \(\frac{1}{4}\) for the first question times \(\frac{1}{4}\) for the second question, resulting in a combined probability of \(\frac{1}{4} * \frac{1}{4} = \frac{1}{16} = 0.0625\) for event A, where both answers are correct.

Complement of an Event

The complement of an event is another fundamental concept in probability; it refers to the event that the original event does not occur. For a given event A, the complement is denoted by A_c and the probability of the event occurring is P(A), while the probability of the complement is P(A_c) = 1 - P(A). It's important to realize that the probability of an event and its complement together add up to 1, which is the certainty that one of them will happen.

In the context of our exercise, the event B is 'at least one answer is correct.' To find the probability of this event, we calculated the probability of its complement, which is the event that both answers are wrong. With individual wrong answer probabilities of \(\frac{3}{4}\), the complement's combined probability is \(\frac{3}{4} * \frac{3}{4} = \frac{9}{16} = 0.5625\). Therefore, the probability of at least one answer being correct, or P(B), is 1 - 0.5625 = 0.4375.