A Bag Contains 10 Red Marbles And 15 Blue Marbles. Suppose 2 Marbles Are Selected From The Bag, Without Replacement. What Is The Probability That 2 Red Marbles Are Selected? Express Your Answer As A Fraction.A. 19/49 B. 4/25 C. 3/20

Feb 28, 2025 by ADMIN 237 views

**A Bag of Marbles: Calculating the Probability of Selecting 2 Red Marbles**

Introduction

In this article, we will explore the concept of probability and how it applies to a simple scenario involving a bag of marbles. We will calculate the probability of selecting 2 red marbles from a bag containing 10 red marbles and 15 blue marbles, without replacement.

Understanding the Problem

The problem states that we have a bag containing 10 red marbles and 15 blue marbles. We are asked to find the probability of selecting 2 red marbles from the bag, without replacement. This means that once a marble is selected, it is not replaced in the bag, and the next marble is selected from the remaining marbles.

Calculating the Probability

To calculate the probability of selecting 2 red marbles, we need to consider the total number of ways to select 2 marbles from the bag, as well as the number of ways to select 2 red marbles.

Step 1: Calculate the Total Number of Ways to Select 2 Marbles

The total number of marbles in the bag is 10 (red) + 15 (blue) = 25. We want to select 2 marbles from the bag, so we need to calculate the number of ways to do this. This can be done using the combination formula:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of marbles (25), k is the number of marbles we want to select (2), and ! denotes the factorial function.

C(25, 2) = 25! / (2!(25-2)!) = 25! / (2!23!) = (25 × 24) / (2 × 1) = 300

So, there are 300 ways to select 2 marbles from the bag.

Step 2: Calculate the Number of Ways to Select 2 Red Marbles

We want to select 2 red marbles from the 10 red marbles in the bag. We can use the combination formula again to calculate the number of ways to do this:

C(10, 2) = 10! / (2!(10-2)!) = 10! / (2!8!) = (10 × 9) / (2 × 1) = 45

So, there are 45 ways to select 2 red marbles from the bag.

Step 3: Calculate the Probability

Now that we have calculated the total number of ways to select 2 marbles (300) and the number of ways to select 2 red marbles (45), we can calculate the probability of selecting 2 red marbles:

P(2 red marbles) = Number of ways to select 2 red marbles / Total number of ways to select 2 marbles = 45 / 300 = 3/20

Conclusion

In this article, we calculated the probability of selecting 2 red marbles from a bag containing 10 red marbles and 15 blue marbles, without replacement. We used the combination formula to calculate the total number of ways to select 2 marbles and the number of ways to select 2 red marbles. The probability of selecting 2 red marbles is 3/20.

Discussion

This problem is a classic example of a probability problem involving combinations. The concept of combinations is used to calculate the number of ways to select a certain number of items from a larger set, without regard to the order in which they are selected.

In this case, we used the combination formula to calculate the total number of ways to select 2 marbles and the number of ways to select 2 red marbles. The probability of selecting 2 red marbles is then calculated by dividing the number of ways to select 2 red marbles by the total number of ways to select 2 marbles.

This problem can be extended to more complex scenarios, such as selecting 3 marbles from a bag containing 10 red marbles and 15 blue marbles, or selecting 2 marbles from a bag containing 5 red marbles and 10 blue marbles.

Solutions to Related Problems

Selecting 3 marbles from a bag containing 10 red marbles and 15 blue marbles, without replacement:

C(25, 3) = 25! / (3!(25-3)!) = 25! / (3!22!) = (25 × 24 × 23) / (3 × 2 × 1) = 2300

C(10, 3) = 10! / (3!(10-3)!) = 10! / (3!7!) = (10 × 9 × 8) / (3 × 2 × 1) = 120

P(3 red marbles) = Number of ways to select 3 red marbles / Total number of ways to select 3 marbles = 120 / 2300 = 6/115
Selecting 2 marbles from a bag containing 5 red marbles and 10 blue marbles, without replacement:

C(15, 2) = 15! / (2!(15-2)!) = 15! / (2!13!) = (15 × 14) / (2 × 1) = 105

C(5, 2) = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4) / (2 × 1) = 10

P(2 red marbles) = Number of ways to select 2 red marbles / Total number of ways to select 2 marbles = 10 / 105 = 2/21
Selecting 2 marbles from a bag containing 8 red marbles and 12 blue marbles, without replacement:

C(20, 2) = 20! / (2!(20-2)!) = 20! / (2!18!) = (20 × 19) / (2 × 1) = 190

C(8, 2) = 8! / (2!(8-2)!) = 8! / (2!6!) = (8 × 7) / (2 × 1) = 28

P(2 red marbles) = Number of ways to select 2 red marbles / Total number of ways to select 2 marbles = 28 / 190 = 14/95
A Bag of Marbles: Calculating the Probability of Selecting 2 Red Marbles ====================================================================================

Q&A: A Bag of Marbles

Q: What is the probability of selecting 2 red marbles from a bag containing 10 red marbles and 15 blue marbles, without replacement?

A: The probability of selecting 2 red marbles is 3/20.

Q: How do you calculate the probability of selecting 2 red marbles?

A: To calculate the probability of selecting 2 red marbles, we need to consider the total number of ways to select 2 marbles from the bag, as well as the number of ways to select 2 red marbles. We can use the combination formula to calculate these values.

Q: What is the combination formula?

A: The combination formula is:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of marbles, k is the number of marbles we want to select, and ! denotes the factorial function.

Q: How do you calculate the total number of ways to select 2 marbles?

A: To calculate the total number of ways to select 2 marbles, we can use the combination formula:

C(25, 2) = 25! / (2!(25-2)!) = 25! / (2!23!) = (25 × 24) / (2 × 1) = 300

Q: How do you calculate the number of ways to select 2 red marbles?

A: To calculate the number of ways to select 2 red marbles, we can use the combination formula:

C(10, 2) = 10! / (2!(10-2)!) = 10! / (2!8!) = (10 × 9) / (2 × 1) = 45

Q: What is the probability of selecting 2 red marbles?

A: The probability of selecting 2 red marbles is:

P(2 red marbles) = Number of ways to select 2 red marbles / Total number of ways to select 2 marbles = 45 / 300 = 3/20

Q: Can you explain the concept of combinations?

A: Yes, the concept of combinations is used to calculate the number of ways to select a certain number of items from a larger set, without regard to the order in which they are selected.

Q: How do you calculate the probability of selecting 3 red marbles from a bag containing 10 red marbles and 15 blue marbles, without replacement?

A: To calculate the probability of selecting 3 red marbles, we need to consider the total number of ways to select 3 marbles from the bag, as well as the number of ways to select 3 red marbles. We can use the combination formula to calculate these values.

Q: What is the total number of ways to select 3 marbles?

A: The total number of ways to select 3 marbles is:

C(25, 3) = 25! / (3!(25-3)!) = 25! / (3!22!) = (25 × 24 × 23) / (3 × 2 × 1) = 2300

Q: What is the number of ways to select 3 red marbles?

A: The number of ways to select 3 red marbles is:

C(10, 3) = 10! / (3!(10-3)!) = 10! / (3!7!) = (10 × 9 × 8) / (3 × 2 × 1) = 120

Q: What is the probability of selecting 3 red marbles?

A: The probability of selecting 3 red marbles is:

P(3 red marbles) = Number of ways to select 3 red marbles / Total number of ways to select 3 marbles = 120 / 2300 = 6/115

Q: Can you explain the concept of probability?

A: Yes, the concept of probability is used to measure the likelihood of an event occurring. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Q: How do you calculate the probability of selecting 2 marbles from a bag containing 5 red marbles and 10 blue marbles, without replacement?

A: To calculate the probability of selecting 2 marbles, we need to consider the total number of ways to select 2 marbles from the bag, as well as the number of ways to select 2 red marbles. We can use the combination formula to calculate these values.

Q: What is the total number of ways to select 2 marbles?

A: The total number of ways to select 2 marbles is:

C(15, 2) = 15! / (2!(15-2)!) = 15! / (2!13!) = (15 × 14) / (2 × 1) = 105

Q: What is the number of ways to select 2 red marbles?

A: The number of ways to select 2 red marbles is:

C(5, 2) = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4) / (2 × 1) = 10

Q: What is the probability of selecting 2 red marbles?

A: The probability of selecting 2 red marbles is:

P(2 red marbles) = Number of ways to select 2 red marbles / Total number of ways to select 2 marbles = 10 / 105 = 2/21

Q: Can you explain the concept of factorial?

A: Yes, the factorial of a number is the product of all positive integers less than or equal to that number. It is denoted by the symbol !. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.