Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A student studying for a vocabulary test knows the meanings of 12 words from a list of 20 words. If the test contains 10 words from the study list, what is the probability that at least 8 of the words on the test are words that the student knows?

Step-by-Step Solution

Answer

The probability that the student knows at least 8 words from the test is calculated by summing the probabilities of three scenarios: knowing 8, 9, or 10 words from the test. To find these probabilities, we use the combination formula: \(C(n,r) = \frac{n!}{r! (n-r)!}\). Probability of knowing at least 8 words = \(\frac{C(12, 8) * C(8, 2)}{C(20, 10)}\) + \(\frac{C(12 ,9) * C(8, 1)}{C(20, 10)}\) + \(\frac{C(12, 10)}{C(20, 10)}\) Calculating the probabilities and summing them up will give us the required probability.

Step 1: Determine the possible scenarios

We are given that there are 20 words in total, out of which the student knows 12 words and the test contains 10 words. We have to find the probability that the student knows at least 8 words from the test. So, the possible scenarios are: the student knows 8, 9, or 10 words from the test.

Step 2: Calculate the combinations for each scenario

In this step, we will calculate the combinations for each of the scenarios mentioned above using the formula nCr, where n is the total number of items, and r is the number of selected items. \(C(n,r) = \frac{n!}{r! (n-r)!}\) Scenario 1: Student knows 8 words from the test. In this case, we need to choose 8 known words and 2 unknown words from the list. Combinations for known words: \(C(12, 8) = \frac{12!}{8! (12-8)!}\) Combinations for unknown words: \(C(8, 2) = \frac{8!}{2! (8-2)!}\) Scenario 2: Student knows 9 words from the test. In this case, we need to choose 9 known words and 1 unknown word from the list. Combinations for known words: \(C(12, 9) = \frac{12!}{9! (12-9)!}\) Combinations for unknown words: \(C(8, 1) = \frac{8!}{1! (8-1)!}\) Scenario 3: Student knows all 10 words from the test. In this case, we need to choose all 10 known words from the list. Combinations for known words: \(C(12, 10) = \frac{12!}{10! (12-10)!}\)

Step 3: Calculate the probability for each scenario

To find the probability for each scenario, we need to divide the product of the combinations of known and unknown words for each scenario by the total possible combinations of selecting 10 words from the list of 20 words. Total combinations of selecting 10 words from the list: \(C(20,10) = \frac{20!}{10! (20-10)!}\) Probability for scenario 1: \(\frac{C(12, 8) * C(8, 2)}{C(20, 10)}\) Probability for scenario 2: \(\frac{C(12 ,9) * C(8, 1)}{C(20, 10)}\) Probability for scenario 3: \(\frac{C(12, 10)}{C(20, 10)}\)

Step 4: Determine the probability of the student knowing at least 8 words

Finally, to find the probability of the student knowing at least 8 words on the test, we will add the probabilities for each of the scenarios mentioned above. Probability of knowing at least 8 words = Probability of knowing 8 words + Probability of knowing 9 words + Probability of knowing 10 words Calculating the probabilities and summing them up will give us the required probability.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting combinations and permutations of objects. It's an essential tool in probability theory to determine the number of possible ways events can occur. In our vocabulary test problem, we use combinatorics to understand the possible ways to select words for the test.

To solve such problems, we typically employ combinations, which help us calculate how many different groups of items we can form from a larger set, without regard to the order of the items. When considering how to select 10 words from a list of 20, for example, combinatorics provides a systematic way to count all the possible choices we have. This is crucial for calculating probabilities related to specific selections, as demonstrated in our example where we need to count how many of these selections satisfy the criteria of including 8 or more words that the student knows.

Binomial Coefficient

The binomial coefficient is represented by the expression \( C(n, r) \) or \( \binom{n}{r} \) and is used to calculate combinations. This notation might also be read as "n choose r" and is integral in finding out how many ways we can pick 'r' items from a total of 'n' items, often used in combinations formula. In our context, we're interested in combinations like selecting 8 known words out of 12 and 2 unknown words out of 8.

The formula for the binomial coefficient is \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n! \) (factorial of n) is the product of all positive integers up to n. This helps us calculate the specific ways in which the student can know exactly 8, 9, or 10 words from the test list. Thus, binomial coefficients are a cornerstone of combinatorial calculations and are crucial in determining probabilities in the exercise.

Mathematical Problem Solving

Mathematical problem solving involves systematically approaching a problem, using logical reasoning and applying mathematical concepts. It helps us break down a problem into manageable steps and find solutions efficiently.

In the vocabulary test exercise, our problem-solving journey began with understanding the scenario. We identified different cases where the student knows at least 8 words out of the test's 10. Each step, such as calculating specific combinations, follows logically from previous steps.

• First, determining scenarios: The student knowing 8, 9, or 10 words.
• Next, calculating combinations: Applying binomial coefficients for each scenario.
• Finally, summing probabilities: Adding probabilities from each scenario to find the final solution.

Each of these steps is a building block that allowed us to solve the problem holistically and reach the solution methodically.