Question

A student placement center has requests from five students for interviews regarding employment with a particular consulting firm. Three of these students are math majors, and the other two students are statistics majors. Unfortunately, the interviewer has time to talk to only two of the students; these two will be randomly selected from among the five. a. What is the probability that both selected students are statistics majors? b. What is the probability that both students are math majors? c. What is the probability that at least one of the students selected is a statistics major? d. What is the probability that the selected students have different majors?

Short answer

a) The probability that both selected students are statistics majors is 0.1. b) The probability that both students are math majors is 0.3. c) The probability that at least one of the students selected is a statistics major is 0.7. d) The probability that the selected students have different majors is 0.6.

Step-by-step solution

Total combinations

First calculate the total number of ways to choose 2 students out of 5. This can be done with the combination formula \( {n \choose k} \). Here, \( n = 5 \) and \( k = 2 \), so the total possible combinations are \({5 \choose 2} = \frac{5!}{2!(5-2)!} = 10\).

Probability of selecting two statistics majors

Now calculate the number of ways to choose 2 students out of the 2 statistics majors, which is \({2 \choose 2} = 1\). The probability is this number divided by the total possible combinations. Therefore, the probability is \(\frac{1}{10} = 0.1\).

Probability of selecting two mathematics majors

Next, calculate the number of ways to choose 2 students out of the 3 mathematics majors, which is \({3 \choose 2} = 3\). Dividing this by the total possible combinations, the probability is \(\frac{3}{10} = 0.3\).

Probability of selecting at least one Statistics major

To calculate this probability, consider the opposite event, which is the event that no Statistics majors are selected. This is same as choosing two students from Mathematics majors. From step 3, we know this probability is 0.3. Therefore, the probability of selecting at least one Statistics major is \(1 - 0.3 = 0.7\).

Probability of selecting students with different majors

The event of choosing students with different majors can happen in two ways: choosing one student from Mathematics major and one student from Statistics, or vice versa. The number of ways to choose one student from Mathematics major and one from Statistics majors is \({3 \choose 1} \cdot {2 \choose 1} = 3 \cdot 2 = 6\). The probability is this number divided by total combinations, giving \(\frac{6}{10} = 0.6\).

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that involves counting, arranging, and grouping objects in specific ways. It plays a crucial role in understanding probability as it helps us quantify the number of possible outcomes in various scenarios. For instance, when we are dealing with a group of students and want to select some of them for an interview, combinatorics provides the tools to calculate all possible selections.

In the given exercise, we use combinatorial methods to count the number of ways to choose two students out of five. This type of problem is generally solved using combinations, denoted by the symbol \( {n \choose k} \), where \( n \) represents the total number of items to choose from, and \( k \) stands for the number of items to choose. The formula for computing combinations is \( {n \choose k} = \frac{n!}{k!(n-k)!} \), where \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).

Understanding combinatorics is not only essential for solving this particular type of problem but for a myriad of probability calculations where outcomes are discrete and their orders do not matter.

Probability Calculations

Probability calculations are at the heart of statistical analysis, allowing us to assign a numerical value to the likelihood of a particular event occurring. In our exercise scenario, we were tasked to determine the probability of selecting specific groups of students, such as two statistics majors or two math majors.

To calculate these probabilities, we divide the number of favorable outcomes (e.g., the number of ways to pick two statistics majors) by the total number of possible outcomes (e.g., the total ways to choose any two students). This fundamental approach to probability is expressed as \( P(E) = \frac{{\text{{number of favorable outcomes}}}}{{\text{{total number of possible outcomes}}}} \), where \( P(E) \) represents the probability of event \( E \) occurring.

For instance, we found that the probability of selecting two statistics majors (a highly specific outcome) was \( \frac{1}{10} \), while the likelihood of selecting two students with different majors (a less specific outcome) was \( \frac{6}{10} \). Arithmetic in probability calculations is often straightforward, but understanding the context and how to count outcomes is key to accurate computation.

Statistics Education

Statistics education is fundamental because it equips students with the necessary skills to interpret data and make informed decisions based on that data. Going beyond memorization of formulas and theorems, statistics education should focus on problem-solving and critical thinking.

In the context of the given exercise, understanding the principles behind the calculations enhances comprehension. For instance, grasping the concept of 'at least one' in probability requires knowledge of complementary events, which in turn can be generalized to tackle a wide range of problems. Moreover, appreciating the different scenarios, such as selecting students with different majors, involves recognizing that multiple pathways can lead to the same outcome, a concept that is widely applicable in statistical problem-solving.

Effective statistics education relies on clear, easy-to-understand explanations and relatable examples to help students internalize the methodologies. By applying the concepts learned to various exercises, as seen with the five students and the interview slots, students can reinforce their knowledge and acquire the confidence to approach more complex statistical challenges.