A Bag Contains 1 Red, 1 Yellow, 1 Blue, And 1 Green Marble. What Is The Probability Of Choosing A Green Marble, Not Replacing It, And Then Choosing A Red Marble?A. $\frac{1}{16}$ B. $\frac{1}{12}$ C. $\frac{1}{4}$ D.

Introduction

Probability is a fundamental concept in mathematics that helps us understand the likelihood of events occurring. In this article, we will explore a classic problem involving a bag of marbles, and we will calculate the probability of choosing a green marble, not replacing it, and then choosing a red marble.

The Problem

A bag contains 1 red, 1 yellow, 1 blue, and 1 green marble. We want to find the probability of choosing a green marble, not replacing it, and then choosing a red marble. This is a classic problem of conditional probability, where the outcome of the first event affects the probability of the second event.

Step 1: Choosing a Green Marble

The probability of choosing a green marble from the bag is simply the number of green marbles divided by the total number of marbles. In this case, there is 1 green marble out of a total of 4 marbles.

Probability of Choosing a Green Marble

• Number of Green Marbles: 1
• Total Number of Marbles: 4
• Probability of Choosing a Green Marble: $\frac{1}{4}$

Step 2: Not Replacing the Green Marble

Since we are not replacing the green marble, the total number of marbles in the bag decreases by 1. Now, there are 3 marbles left in the bag, and we want to find the probability of choosing a red marble from this reduced set.

Probability of Choosing a Red Marble

• Number of Red Marbles: 1
• Total Number of Marbles: 3
• Probability of Choosing a Red Marble: $\frac{1}{3}$

Step 3: Choosing a Red Marble

Now, we want to find the probability of choosing a red marble after choosing a green marble and not replacing it. This is a conditional probability problem, where the outcome of the first event affects the probability of the second event.

Conditional Probability

The conditional probability of choosing a red marble after choosing a green marble and not replacing it is given by the product of the probabilities of the two events.

Conditional Probability of Choosing a Red Marble

• Probability of Choosing a Green Marble: $\frac{1}{4}$
• Probability of Choosing a Red Marble: $\frac{1}{3}$
• Conditional Probability of Choosing a Red Marble: $\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$

Conclusion

In this article, we explored a classic problem involving a bag of marbles and calculated the probability of choosing a green marble, not replacing it, and then choosing a red marble. We found that the conditional probability of choosing a red marble after choosing a green marble and not replacing it is $\frac{1}{12}$.

Key Takeaways

• The probability of choosing a green marble from a bag of 4 marbles is $\frac{1}{4}$.
• The probability of choosing a red marble from a bag of 3 marbles is $\frac{1}{3}$.
• The conditional probability of choosing a red marble after choosing a green marble and not replacing it is $\frac{1}{12}$.

Frequently Asked Questions

Q: What is the probability of choosing a green marble from a bag of 4 marbles?

A: The probability of choosing a green marble from a bag of 4 marbles is $\frac{1}{4}$.

Q: What is the probability of choosing a red marble from a bag of 3 marbles?

A: The probability of choosing a red marble from a bag of 3 marbles is $\frac{1}{3}$.

Q: What is the conditional probability of choosing a red marble after choosing a green marble and not replacing it?

A: The conditional probability of choosing a red marble after choosing a green marble and not replacing it is $\frac{1}{12}$.

References

• Probability Theory
• Conditional Probability
• Marbles and Probability

Further Reading

• Probability and Statistics
• Conditional Probability and Independence
• Marbles and Probability Problems
  A Bag of Marbles: Understanding Probability - Q&A =====================================================

Introduction

In our previous article, we explored a classic problem involving a bag of marbles and calculated the probability of choosing a green marble, not replacing it, and then choosing a red marble. We found that the conditional probability of choosing a red marble after choosing a green marble and not replacing it is $\frac{1}{12}$. In this article, we will answer some frequently asked questions related to this problem.

Q&A

Q: What is the probability of choosing a green marble from a bag of 4 marbles?

A: The probability of choosing a green marble from a bag of 4 marbles is $\frac{1}{4}$.

Q: What is the probability of choosing a red marble from a bag of 3 marbles?

A: The probability of choosing a red marble from a bag of 3 marbles is $\frac{1}{3}$.

Q: What is the conditional probability of choosing a red marble after choosing a green marble and not replacing it?

A: The conditional probability of choosing a red marble after choosing a green marble and not replacing it is $\frac{1}{12}$.

Q: What if we replace the green marble after choosing it?

A: If we replace the green marble after choosing it, the total number of marbles in the bag remains the same. In this case, the probability of choosing a red marble is still $\frac{1}{4}$.

Q: What if we choose a red marble first and then a green marble?

A: If we choose a red marble first and then a green marble, the probability of choosing a green marble is $\frac{1}{3}$, since there are only 3 marbles left in the bag.

Q: Can we use the formula for conditional probability to solve this problem?

A: Yes, we can use the formula for conditional probability to solve this problem. The formula is:

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

In this case, A is the event of choosing a red marble, and B is the event of choosing a green marble.

Q: What is the probability of choosing a green marble and then a red marble?

A: The probability of choosing a green marble and then a red marble is the product of the probabilities of the two events:

P(G and R) = P(G) x P(R|G) = $\frac{1}{4}$ x $\frac{1}{3}$ = $\frac{1}{12}$

Q: What is the probability of choosing a red marble and then a green marble?

A: The probability of choosing a red marble and then a green marble is the product of the probabilities of the two events:

P(R and G) = P(R) x P(G|R) = $\frac{1}{4}$ x $\frac{1}{3}$ = $\frac{1}{12}$

Conclusion

In this article, we answered some frequently asked questions related to the problem of choosing a green marble, not replacing it, and then choosing a red marble. We found that the conditional probability of choosing a red marble after choosing a green marble and not replacing it is $\frac{1}{12}$. We also used the formula for conditional probability to solve this problem and found that the probability of choosing a green marble and then a red marble is also $\frac{1}{12}$.

Key Takeaways

• The probability of choosing a green marble from a bag of 4 marbles is $\frac{1}{4}$.
• The probability of choosing a red marble from a bag of 3 marbles is $\frac{1}{3}$.
• The conditional probability of choosing a red marble after choosing a green marble and not replacing it is $\frac{1}{12}$.
• The probability of choosing a green marble and then a red marble is $\frac{1}{12}$.
• The probability of choosing a red marble and then a green marble is $\frac{1}{12}$.

Frequently Asked Questions

Q: What is the probability of choosing a green marble from a bag of 4 marbles?

A: The probability of choosing a green marble from a bag of 4 marbles is $\frac{1}{4}$.

Q: What is the probability of choosing a red marble from a bag of 3 marbles?

A: The probability of choosing a red marble from a bag of 3 marbles is $\frac{1}{3}$.

Q: What is the conditional probability of choosing a red marble after choosing a green marble and not replacing it?

A: The conditional probability of choosing a red marble after choosing a green marble and not replacing it is $\frac{1}{12}$.

Q: What if we replace the green marble after choosing it?

A: If we replace the green marble after choosing it, the total number of marbles in the bag remains the same. In this case, the probability of choosing a red marble is still $\frac{1}{4}$.

Q: What if we choose a red marble first and then a green marble?

A: If we choose a red marble first and then a green marble, the probability of choosing a green marble is $\frac{1}{3}$, since there are only 3 marbles left in the bag.

Q: Can we use the formula for conditional probability to solve this problem?

A: Yes, we can use the formula for conditional probability to solve this problem. The formula is:

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

In this case, A is the event of choosing a red marble, and B is the event of choosing a green marble.

Q: What is the probability of choosing a green marble and then a red marble?

A: The probability of choosing a green marble and then a red marble is the product of the probabilities of the two events:

P(G and R) = P(G) x P(R|G) = $\frac{1}{4}$ x $\frac{1}{3}$ = $\frac{1}{12}$

Q: What is the probability of choosing a red marble and then a green marble?

A: The probability of choosing a red marble and then a green marble is the product of the probabilities of the two events:

P(R and G) = P(R) x P(G|R) = $\frac{1}{4}$ x $\frac{1}{3}$ = $\frac{1}{12}$

References

• Probability Theory
• Conditional Probability
• Marbles and Probability

Further Reading

• Probability and Statistics
• Conditional Probability and Independence
• Marbles and Probability Problems