Question

A student prepares for an exam by studying a list of 10 problems. She can solve 6 of them. For the exam, the instructor selects 5 problems at random from the list of \(10 .\) What is the probability that the student can solve all 5 problems on the exam?

Short answer

Answer: The probability that the student can solve all 5 problems on the exam is 1/42.

Step-by-step solution

Find the total number of combinations

The instructor selects 5 problems from the list of 10. To find the total number of possible combinations, we use the "combination formula": \(\textit{Total combinations} = \binom{n}{r}\), where \(n=10\) (problems in the list) and \(r=5\) (problems selected for the exam). \(\textit{Total combinations} = \binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!}\) Now, let's calculate the value: \(\textit{Total combinations} = \frac{10\times9\times8\times7\times6}{5\times4\times3\times2\times1} = 252\) So, there are 252 total possible combinations.

Find the favorable combinations

The student can solve 6 problems from the list of 10. To find the favorable combinations, the instructor must select 5 problems from these 6 problems that the student knows. We can use the combination formula again: \(\textit{Favorable combinations} = \binom{n}{r}\), where \(n=6\) (problems she knows) and \(r=5\) (problems selected for the exam). \(\textit{Favorable combinations} = \binom{6}{5} = \frac{6!}{5!(6-5)!} = \frac{6!}{5!1!}\) Now, let's calculate the value: \(\textit{Favorable combinations} = \frac{6\times5\times4\times3\times2}{5\times4\times3\times2\times1} = 6\) So, there are 6 favorable combinations.

Calculate the probability

Now, we have the total combinations and the favorable combinations. We can find the probability using the formula: \(\textit{Probability} = \frac{\textit{Favorable combinations}}{\textit{Total combinations}}\) Substitute the values and calculate the probability: \(\textit{Probability} = \frac{6}{252} = \frac{1}{42}\) Hence, the probability that the student can solve all 5 problems on the exam is \(\frac{1}{42}\).

Key concepts

Combination Formula

Understanding the combination formula is crucial in evaluating the number of ways a given set of elements can be selected. The formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) represents the number of combinations when choosing \(r\) elements from a larger set of \(n\) elements where order doesn't matter.

In our exercise, this formula is used twice to calculate distinct scenarios. Initially, to determine the total number of ways that 5 problems can be chosen out of 10, we use \(n = 10\) and \(r = 5\). This calculation finds all the possible selections of questions the instructor could make. Next, we consider only the subset of problems that the student can solve, with \(n = 6\) and \(r = 5\), to compute the favorable outcomes.

Importance of the Combination Formula

• It ignores the order, focusing on selection rather than arrangement, which is useful in many real-life scenarios.
• It helps to simplify complex probability problems into manageable calculations.
• It lays the foundation for understanding more advanced topics in statistics and probability.

Combinatorics in Probability

Combinatorics is the branch of mathematics dealing with combinations, permutations, and their application in various fields, including probability. Probability often involves calculating the likelihood of certain combinations, as showcased in the given problem.

By applying combinatorial analysis, we assess possible outcomes and their occurrence chances. Here, we first ascertain all the ways to choose 5 questions from the total set of 10, and then, from the subset of 6 questions the student can solve, again determine the ways to choose 5 questions.

Role in Probability

• Combinatorics allows for precise calculation of event spaces in probability.
• It aids in distinguishing between 'favorable' and 'possible' outcomes to calculate probabilities accurately.

The alignment of combinatorics and probability is essential for problems where the outcome set is large and counting individual possibilities is impractical.

Calculating Probabilities

Probability is the measure of the likelihood that an event will occur, symbolized as a number between 0 and 1. The probability formula \(\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total possible outcomes}}\) guides us to quantify this likelihood.

In our exercise, this formula is applied to calculate the probability that the student can solve all 5 problems chosen for the exam. With 6 favorable outcomes (combinations of questions the student can solve) and 252 total possible outcomes (combinations of any 5 questions), the probability is calculated as \(\frac{6}{252} = \frac{1}{42}\). It tells us that the student has a 1 in 42 chance of being able to solve all chosen problems on the exam.

Understanding Probability Calculations

• It's instrumental in determining the odds of various outcomes, influencing decision-making and risk analysis.
• Probabilities can serve as a predictive tool in a wide range of disciplines, from statistics to game theory and finance.

The mastery of probability calculations paves the way for students to confidently tackle a wide spectrum of statistical problems.