Problem 32 - Algebra 2

Question

A student guesses at all 5 questions on a true-false quiz. Find each probability. \(P(\text { exactly } 4 \text { correct })\)

Step-by-Step Solution

Answer

The probability is \( \frac{5}{32} \).

Step 1: Define the Parameters of a True-False Quiz

Each question in a true-false quiz has two possible answers: True or False. Let's define the probability of a student guessing a question correctly as \( p = \frac{1}{2} \). The quiz consists of 5 questions, so \( n = 5 \) represents the number of trials.

Step 2: Understand the Probability of Success

To find the probability of getting exactly 4 correct answers, we need to use the binomial probability formula, which is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( k \) is the number of successes (correct answers) we are interested in.

Step 3: Calculate the Binomial Coefficient

Calculate the binomial coefficient for 4 correct answers out of 5 questions: \( \binom{5}{4} = \frac{5!}{4!(5-4)!} = 5 \). This tells us there are 5 ways to choose 4 correct answers out of 5 questions.

Step 4: Apply the Binomial Probability Formula

Substitute the values into the binomial probability formula: \( P(\text{exactly 4 correct}) = \binom{5}{4} \left(\frac{1}{2}\right)^4 \left(1-\frac{1}{2}\right)^{5-4} = 5 \cdot \left(\frac{1}{16}\right) \cdot \left(\frac{1}{2}\right) = \frac{5}{32} \).

Step 5: Interpret the Result

The probability of the student guessing exactly 4 correct answers on the quiz is \( \frac{5}{32} \). This is approximately 0.15625 when converted to a decimal.

Key Concepts

True-False Quiz

A true-false quiz is a type of test where each question has only two possible answers: True or False. This makes it unique because, when taking such a quiz, each question has a 50% chance of being answered correctly if answered at random. Understanding this concept is crucial when dealing with probability problems related to such quizzes.

• Each question is a separate event, so the probability for each question remains constant independent of previous questions.
• If a test consists of multiple questions, like a 5-question quiz, a participant has multiple opportunities (5 in total) to answer correctly.

In our original exercise, we consider a scenario where a student guesses the answers randomly, meaning they rely entirely on this 50% chance for each question.

Probability of Success

The probability of success in a given experiment refers to the likelihood of a favorable outcome—in our case, a correct answer. For a true-false quiz, each question has a probability of correct guessing, or success, which is \( p = \frac{1}{2} \). This means there's an equal chance of answering correctly or incorrectly.
If you want to know the probability of achieving exactly 4 correct answers out of 5 questions, you're focusing on one specific outcome among many possibilities:

• The successes here are the correct answers.
• The probability formula used requires identifying the number of successful results we're interested in (4 correct answers).

This understanding helps prepare you for using the full binomial probability formula to calculate the exact success probability.

Binomial Coefficient

The binomial coefficient is a fundamental part of calculating binomial probabilities. It represents the number of different combinations or ways you can choose a certain number of successes (correct answers) from a larger set of trials (questions). For instance, in the original problem:

• The binomial coefficient \( \binom{5}{4} \) answers "How many ways can you choose 4 questions to get right out of 5?"
• This is calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of questions and \( k \) is the number of chosen successes.

Understanding this is necessary to apply the binomial formula. For our example, there are 5 ways to choose which of the 5 questions will be answered correctly.

Combination Formula

The combination formula, often presented using the binomial coefficient symbol \( \binom{n}{k} \), calculates how many ways you can choose a subset of items from a larger set without regard to order. It's key for binomial probabilities, since it helps determine the different ways successes can occur in a series of trials.

• In notation, \( n \) is the total number of items, and \( k \) is the number of items to choose.
• The formula is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Factorials, denoted by an exclamation mark (!), mean multiplying a series of descending numbers. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). This concept ensures you accurately determine the combinations needed for solving a probability problem like the one discussed. If correctly applied, the combination formula helps compute probabilities in binomial distributions efficiently and correctly.