A Problem Is Given To Two Students, A And B, Whose Chances Of Solving It Are 1 3 \frac{1}{3} 3 1 ​ And 1 4 \frac{1}{4} 4 1 ​ Respectively. What Is The Probability That The Problem Will Be Solved?

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Introduction

In probability theory, the chances of an event occurring are often represented as a fraction or a decimal value between 0 and 1. When two or more events are independent, the probability of both events occurring can be calculated by multiplying their individual probabilities. However, when the events are dependent, the probability of both events occurring is more complex to calculate. In this article, we will explore the concept of probability and how to calculate the probability that a problem will be solved by two students, A and B, whose chances of solving it are $\frac{1}{3}$ and $\frac{1}{4}$ respectively.

Understanding Probability

Probability is a measure of the likelihood of an event occurring. It is often represented as a fraction or a decimal value between 0 and 1. The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if a coin is flipped, the probability of it landing heads up is $\frac{1}{2}$, because there are two possible outcomes (heads or tails) and only one of them is favorable.

Calculating the Probability of Two Independent Events

When two events are independent, the probability of both events occurring can be calculated by multiplying their individual probabilities. This is because the occurrence of one event does not affect the probability of the other event occurring. For example, if the probability of a student, A, solving a problem is $\frac{1}{3}$ and the probability of a student, B, solving the same problem is $\frac{1}{4}$, the probability that both students will solve the problem is $\frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$.

Calculating the Probability of Two Dependent Events

However, when the events are dependent, the probability of both events occurring is more complex to calculate. Dependent events are events where the occurrence of one event affects the probability of the other event occurring. For example, if a student, A, solves a problem, the probability of a student, B, solving the same problem may increase or decrease. In this case, the probability of both students solving the problem cannot be calculated by simply multiplying their individual probabilities.

The Problem of Two Students, A and B

In this problem, we are given that the chances of student A solving the problem are $\frac{1}{3}$ and the chances of student B solving the problem are $\frac{1}{4}$. We are asked to find the probability that the problem will be solved. Since the problem does not specify whether the events are independent or dependent, we will assume that the events are independent.

Calculating the Probability of the Problem Being Solved

To calculate the probability of the problem being solved, we need to consider two possible scenarios:

• Student A solves the problem, and student B does not solve the problem.
• Student A does not solve the problem, and student B solves the problem.

The probability of student A solving the problem is $\frac{1}{3}$, and the probability of student B not solving the problem is $1 - \frac{1}{4} = \frac{3}{4}$. Therefore, the probability of student A solving the problem and student B not solving the problem is $\frac{1}{3} \times \frac{3}{4} = \frac{1}{4}$.

Similarly, the probability of student A not solving the problem is $1 - \frac{1}{3} = \frac{2}{3}$, and the probability of student B solving the problem is $\frac{1}{4}$. Therefore, the probability of student A not solving the problem and student B solving the problem is $\frac{2}{3} \times \frac{1}{4} = \frac{1}{6}$.

Adding the Probabilities

To find the total probability of the problem being solved, we need to add the probabilities of the two scenarios:

$\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$

Therefore, the probability that the problem will be solved is $\frac{5}{12}$.

Conclusion

In this article, we explored the concept of probability and how to calculate the probability that a problem will be solved by two students, A and B, whose chances of solving it are $\frac{1}{3}$ and $\frac{1}{4}$ respectively. We assumed that the events are independent and calculated the probability of the problem being solved by considering two possible scenarios. The total probability of the problem being solved is $\frac{5}{12}$.

References

• Probability Theory
• Independent Events
• Dependent Events

Further Reading

• Probability and Statistics
• Mathematical Probability
• Probability Theory and Applications
  

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Q&A: A Problem is Given to Two Students, A and B, Whose Chances of Solving It Are $\frac{1}{3}$ and $\frac{1}{4}$ Respectively. What is the Probability That the Problem Will Be Solved?

Q: What is the probability that student A will solve the problem?

A: The probability that student A will solve the problem is $\frac{1}{3}$.

Q: What is the probability that student B will solve the problem?

A: The probability that student B will solve the problem is $\frac{1}{4}$.

Q: What is the probability that the problem will be solved if student A solves it and student B does not solve it?

A: The probability that student A solves the problem is $\frac{1}{3}$, and the probability that student B does not solve the problem is $\frac{3}{4}$. Therefore, the probability that student A solves the problem and student B does not solve it is $\frac{1}{3} \times \frac{3}{4} = \frac{1}{4}$.

Q: What is the probability that the problem will be solved if student A does not solve it and student B solves it?

A: The probability that student A does not solve the problem is $\frac{2}{3}$, and the probability that student B solves the problem is $\frac{1}{4}$. Therefore, the probability that student A does not solve the problem and student B solves it is $\frac{2}{3} \times \frac{1}{4} = \frac{1}{6}$.

Q: What is the total probability that the problem will be solved?

A: To find the total probability that the problem will be solved, we need to add the probabilities of the two scenarios:

$\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$

Therefore, the total probability that the problem will be solved is $\frac{5}{12}$.

Q: What if the events are dependent?

A: If the events are dependent, the probability of the problem being solved is more complex to calculate. We would need to consider the probability of student A solving the problem given that student B solves the problem, and vice versa.

Q: How do we calculate the probability of dependent events?

A: To calculate the probability of dependent events, we need to use conditional probability. This involves calculating the probability of one event occurring given that another event has occurred.

Q: What is conditional probability?

A: Conditional probability is a measure of the probability of an event occurring given that another event has occurred. It is calculated by dividing the probability of the two events occurring by the probability of the first event occurring.

Q: How do we use conditional probability to calculate the probability of dependent events?

A: To use conditional probability to calculate the probability of dependent events, we need to calculate the probability of one event occurring given that another event has occurred. This involves dividing the probability of the two events occurring by the probability of the first event occurring.

Q: What is the formula for conditional probability?

A: The formula for conditional probability is:

P(A|B) = P(A and B) / P(B)

Where P(A|B) is the probability of event A occurring given that event B has occurred, P(A and B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.

Q: How do we apply the formula for conditional probability to the problem of two students, A and B?

A: To apply the formula for conditional probability to the problem of two students, A and B, we need to calculate the probability of student A solving the problem given that student B solves the problem, and vice versa.

Q: What is the probability that student A solves the problem given that student B solves the problem?

A: To calculate the probability that student A solves the problem given that student B solves the problem, we need to use the formula for conditional probability:

P(A|B) = P(A and B) / P(B)

Where P(A|B) is the probability of student A solving the problem given that student B solves the problem, P(A and B) is the probability of both students A and B solving the problem, and P(B) is the probability of student B solving the problem.

Q: What is the probability that student B solves the problem given that student A solves the problem?

A: To calculate the probability that student B solves the problem given that student A solves the problem, we need to use the formula for conditional probability:

P(B|A) = P(A and B) / P(A)

Where P(B|A) is the probability of student B solving the problem given that student A solves the problem, P(A and B) is the probability of both students A and B solving the problem, and P(A) is the probability of student A solving the problem.

Q: How do we calculate the probability of both students A and B solving the problem?

A: To calculate the probability of both students A and B solving the problem, we need to multiply the probabilities of each student solving the problem:

P(A and B) = P(A) * P(B)

Where P(A and B) is the probability of both students A and B solving the problem, P(A) is the probability of student A solving the problem, and P(B) is the probability of student B solving the problem.

Q: What is the probability that both students A and B solve the problem?

A: To calculate the probability that both students A and B solve the problem, we need to multiply the probabilities of each student solving the problem:

P(A and B) = P(A) * P(B)

Where P(A and B) is the probability of both students A and B solving the problem, P(A) is the probability of student A solving the problem, and P(B) is the probability of student B solving the problem.

Q: What is the probability that the problem will be solved if both students A and B solve it?

A: To calculate the probability that the problem will be solved if both students A and B solve it, we need to multiply the probabilities of each student solving the problem:

P(A and B) = P(A) * P(B)

Where P(A and B) is the probability of both students A and B solving the problem, P(A) is the probability of student A solving the problem, and P(B) is the probability of student B solving the problem.

Q: What is the total probability that the problem will be solved?

A: To find the total probability that the problem will be solved, we need to add the probabilities of the two scenarios:

$\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$

Therefore, the total probability that the problem will be solved is $\frac{5}{12}$.

Q: What if the events are dependent and the probability of student A solving the problem given that student B solves the problem is $\frac{2}{3}$?

A: If the events are dependent and the probability of student A solving the problem given that student B solves the problem is $\frac{2}{3}$, we need to use the formula for conditional probability to calculate the probability of student A solving the problem given that student B solves the problem:

P(A|B) = P(A and B) / P(B)

Where P(A|B) is the probability of student A solving the problem given that student B solves the problem, P(A and B) is the probability of both students A and B solving the problem, and P(B) is the probability of student B solving the problem.

Q: What is the probability that student A solves the problem given that student B solves the problem?

A: To calculate the probability that student A solves the problem given that student B solves the problem, we need to use the formula for conditional probability:

P(A|B) = P(A and B) / P(B)

Where P(A|B) is the probability of student A solving the problem given that student B solves the problem, P(A and B) is the probability of both students A and B solving the problem, and P(B) is the probability of student B solving the problem.

Q: What is the probability that the problem will be solved if student A solves it and student B does not solve it?

A: The probability that student A solves the problem is $\frac{1}{3}$, and the probability that student B does not solve the problem is $\frac{3}{4}$. Therefore, the probability that student A solves the problem and student B does not solve the problem is $\frac{1}{3} \times \frac{3}{4} = \frac{1}{4}$.

Q: What is the probability that the problem will be solved if student A does not solve it and student B solves it?

A: The probability that student A does not solve the problem is $\frac{2}{3}$, and the probability that student B solves the problem is $\frac{1}{4}$. Therefore, the probability that student A does not solve the problem and student B solves it is $\frac{2}{3} \times \frac{1}{4} = \frac{1}{6}$.

Q: What is the total probability that the problem will be