Probability Problem Solving Probability Question By CA Students A B And C
In the realm of probability, we often encounter scenarios where multiple individuals or entities attempt to solve a problem independently. This article delves into a classic probability problem involving three CA (Chartered Accountancy) students, A, B, and C, each with varying chances of solving a particular question. Our goal is to determine the overall probability that the problem will be solved, considering the individual probabilities of success for each student. Let's break down the problem, explore the underlying concepts, and arrive at the solution in a comprehensive manner. We will also explore some ways to apply probability in the real world.
The problem at hand presents a scenario where three CA students, A, B, and C, are given a probability question to solve. Their individual probabilities of solving the problem are given as follows:
- Student A: Probability of solving the problem = 1/3
- Student B: Probability of solving the problem = 1/5
- Student C: Probability of solving the problem = 1/2
The core question we aim to answer is: What is the probability that the problem will be solved, considering the combined efforts of the three students? This requires us to consider the scenarios where at least one of them solves the problem.
Before diving into the solution, it's crucial to grasp the fundamental concepts of probability that underpin this problem. These concepts include:
- Probability of an Event: The probability of an event is a measure of the likelihood that the event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In this case, each student has a probability of solving the problem, representing their individual chances of success.
- Independent Events: Events are considered independent if the outcome of one event does not affect the outcome of the other. In this problem, we assume that the students attempt to solve the problem independently, meaning that one student's success or failure does not influence the others.
- Probability of the Complement: The complement of an event is the event not occurring. The probability of the complement of an event A is denoted as P(A') and is calculated as 1 - P(A), where P(A) is the probability of event A occurring. This concept will be useful in our solution.
- Probability of Union of Events: The union of events refers to the occurrence of at least one of the events. In this problem, we are interested in the probability that at least one student solves the problem, which is the union of the events of each student solving the problem.
To find the probability that the problem will be solved, we can use the concept of the complement. Instead of directly calculating the probability that at least one student solves the problem, we can calculate the probability that none of the students solve the problem and subtract it from 1. This is because:
Probability (Problem solved) = 1 - Probability (Problem not solved by anyone)
Let's break down the steps:
- Calculate the probability that each student does NOT solve the problem:
- P(A does not solve) = 1 - P(A solves) = 1 - (1/3) = 2/3
- P(B does not solve) = 1 - P(B solves) = 1 - (1/5) = 4/5
- P(C does not solve) = 1 - P(C solves) = 1 - (1/2) = 1/2
- Calculate the probability that NONE of the students solve the problem: Since the events are independent, we can multiply the probabilities: P(None solve) = P(A does not solve) * P(B does not solve) * P(C does not solve) P(None solve) = (2/3) * (4/5) * (1/2) = 8/30 = 4/15
- Calculate the probability that the problem IS solved: P(Problem solved) = 1 - P(None solve) P(Problem solved) = 1 - (4/15) = 11/15
Therefore, the probability that the problem would be solved is 11/15.
To further clarify the solution, let's walk through the calculation steps in a more detailed manner. This will help solidify your understanding of the process and the underlying logic.
Step 1: Probability of Each Student NOT Solving the Problem
As we established earlier, the probability of the complement of an event is calculated by subtracting the probability of the event from 1. This gives us the probability that the event will not occur.
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Student A: The probability of student A solving the problem is 1/3. Therefore, the probability of student A not solving the problem is: P(A does not solve) = 1 - P(A solves) = 1 - (1/3) = 2/3
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Student B: Similarly, the probability of student B solving the problem is 1/5. The probability of student B not solving the problem is: P(B does not solve) = 1 - P(B solves) = 1 - (1/5) = 4/5
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Student C: The probability of student C solving the problem is 1/2. The probability of student C not solving the problem is: P(C does not solve) = 1 - P(C solves) = 1 - (1/2) = 1/2
Step 2: Probability of NONE of the Students Solving the Problem
Since we are assuming that the students are working independently, the probability of all three students failing to solve the problem is the product of their individual probabilities of failure. This is because the events are independent, and the outcome of one student's attempt does not influence the others.
Therefore, the probability that none of the students solve the problem is:
P(None solve) = P(A does not solve) * P(B does not solve) * P(C does not solve)
Substituting the values we calculated in Step 1, we get:
P(None solve) = (2/3) * (4/5) * (1/2) = 8/30 = 4/15
This means there is a 4/15 chance that all three students will fail to solve the problem.
Step 3: Probability of the Problem Being Solved
Now that we know the probability that none of the students solve the problem, we can use the complement rule to find the probability that the problem is solved. This is the probability that at least one student solves the problem.
As we established earlier:
Probability (Problem solved) = 1 - Probability (Problem not solved by anyone)
In our case, this translates to:
P(Problem solved) = 1 - P(None solve)
Substituting the value we calculated in Step 2, we get:
P(Problem solved) = 1 - (4/15) = 11/15
Therefore, the probability that the problem would be solved is 11/15. This is the final answer to the problem.
While the complement approach is often the most efficient for this type of problem, it's also possible to calculate the probability directly. This involves considering all the possible scenarios where the problem is solved:
- Only A solves the problem
- Only B solves the problem
- Only C solves the problem
- A and B solve the problem, but C doesn't
- A and C solve the problem, but B doesn't
- B and C solve the problem, but A doesn't
- All three solve the problem
Calculating the probability of each scenario and summing them up will give the same result (11/15), but it's a more lengthy and potentially error-prone process. This highlights the elegance and efficiency of the complement approach.
In conclusion, by applying the principles of probability and the concept of complements, we have successfully determined the probability that the problem would be solved by at least one of the three CA students. The final answer is 11/15, which represents the likelihood of the problem being solved through the combined efforts of students A, B, and C. The detailed step-by-step solution, along with the alternative approach, provides a comprehensive understanding of the problem-solving process. This exercise underscores the power of probability in analyzing and predicting outcomes in situations involving multiple independent events.
- Risk Assessment: Insurance companies use probability to assess the risk of insuring individuals or assets. They analyze historical data and statistical models to determine the likelihood of events such as accidents, illnesses, or natural disasters. This allows them to set premiums and manage their financial exposure.
- Financial Modeling: Investors and financial analysts use probability to model financial markets and make investment decisions. They may use statistical techniques to analyze historical stock prices, interest rates, and economic indicators to estimate the probability of future market movements. This information can help them make informed decisions about buying, selling, or holding assets.
- Medical Decision Making: Doctors and other healthcare professionals use probability to diagnose and treat medical conditions. They may use statistical models to estimate the likelihood of a patient having a particular disease based on their symptoms and test results. They also use probability to assess the effectiveness of different treatments and make decisions about patient care.
- Quality Control: Manufacturers use probability to monitor the quality of their products and processes. They may use statistical techniques to sample products and test them for defects. The results of these tests can be used to estimate the probability of producing defective items and make adjustments to the manufacturing process to improve quality.
- Weather Forecasting: Meteorologists use probability to forecast the weather. They analyze weather data, such as temperature, humidity, and wind speed, and use statistical models to estimate the probability of different weather events, such as rain, snow, or thunderstorms. This information is used to create weather forecasts and warnings.
- Game Theory: Game theory is a branch of mathematics that uses probability to analyze strategic interactions between individuals or groups. It has applications in economics, politics, and other fields. For example, game theory can be used to model the behavior of firms in a competitive market or the negotiations between countries.
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