Question

Determine whether the events are independent. (See Examples I and 2.) A student is taking a multiple-choice test where each question has four choices. The student randomly guesses the answers to the five-question test. Use a sample space to determine whether guessing Question 1 correctly and Question 2 correctly are independent events.

Short answer

Yes, guessing the first question correctly and guessing the second question correctly are independent events. The probability of both events occurring is 0.0625 and this matches the criteria for independent events. The occurrence of one event does not affect the probability of the other event occurring.

Step-by-step solution

Understand the Concept of Independent Events

Two events are independent if the probability of both events occurring is the product of their individual probabilities. In algebraic terms, Events A and B are independent if \(P(A \cap B) = P(A) * P(B)\).

Identify the Probabilities

The student has a 1 out of 4 chance of guessing each question correctly - because each question has 4 choices. So for both question 1 and 2, the probability of the student guessing correctly is \(1/4\) or 0.25.

Apply the Independent Events Rule

We have previously determined that two events A and B are independent if \(P(A \cap B) = P(A) * P(B)\). Applying this to our situation: the probability of guessing both questions correctly \(P(Q1 \cap Q2)\) equals the product of the probability of guessing each question correctly. In mathematical terms: \(P(Q1 \cap Q2) = P(Q1) * P(Q2)\). Substituting the known values, P(Q1)=P(Q2)=0.25, we get \(P(Q1 \cap Q2) = 0.25 * 0.25 = 0.0625\). Therefore, the events under question are indeed independent.

Final Probability Verification

The computed probability for guessing both questions correctly is 0.0625. This value represents the probability that both events (guessing Question 1 correctly and guessing Question 2 correctly) occur together. If we were to verify this in the sample space, we would note that there are 16 possible outcome pairs (as there are 4 possible outcomes for each question and 2 questions). Only 1 of these pairs represents the event that both questions are guessed correctly. So, the probability of both events occurring is \(1/16 = 0.0625\), which matches our earlier computation.

Key concepts

Probability

Probability is a way to measure how likely an event is to happen. In simpler terms, it tells us the chances of something occurring. In mathematics, probability is expressed as a number between 0 and 1, with 0 being impossible and 1 being certain. For example, when you flip a coin, there is a probability of 0.5 (or 50%) that it will land on heads.

The formula for probability is given by the ratio of the favorable outcomes to the total number of possible outcomes. If you have an event with a single favorable outcome like guessing the correct answer to a multiple-choice question among four options, the probability is calculated as:

\[ P = \frac{1}{4} \].

Understanding probability helps in making predictions about how likely various outcomes are, just like determining if events such as guessing answers on a test are independent of one another.

Sample Space

Sample space is a concept used in probability to represent all possible outcomes of a particular experiment or situation. Think of it as a list that includes every single outcome that could happen. For example, if you roll a six-sided die, the sample space consists of the numbers 1 through 6.

In the context of a multiple-choice test, where each question has four possible answers, the sample space for two questions would involve pairs of answers for each question. Thus, if there are two questions, each with four options, the sample space would contain:

• (A1, A1), (A1, A2), (A1, A3), (A1, A4)
• (A2, A1), (A2, A2), (A2, A3), (A2, A4)
• (A3, A1), (A3, A2), (A3, A3), (A3, A4)
• (A4, A1), (A4, A2), (A4, A3), (A4, A4)

This results in a sample space with 16 possible pairs of answers because you have four options for the first question and four for the second, which multiplies to 4 x 4 = 16. Analyzing the sample space allows us to better understand probability and independence between events.

Multiple-Choice Questions

Multiple-choice questions are a common type of question used in exams to test a student's knowledge. Typically, each question presents several possible answers, but only one is correct. In our exercise, each question has four options, giving students a probability of guessing the correct answer as \( \frac{1}{4} \) or 0.25.

When dealing with multiple-choice questions, particularly in probability exercises, it's crucial to consider whether guesses are made independently. In this context, independent events mean that the outcome of one does not affect the outcome of the other.

For instance, if a student guesses on two separate questions, guessing correctly on Question 1 should not influence their chances of guessing correctly on Question 2. This is what it means for the two events (each question being guessed correctly) to be independent. Evaluating such scenarios helps us in understanding larger concepts of probability and independence and can better prepare students for tackling related probability problems.