Calculating Probability of Drawing Specific Marbles from a Bag with Replacement

When working with probability problems that involve drawing items from a bag, the calculations can become a bit complex, especially when considering replacement. This article explores the probability of drawing a blue and then a red marble from a bag containing 3 green, 11 red, and 6 blue marbles. Specifically, we will delve into the process of calculating these probabilities and understand the underlying concepts of independent events and replacement.

Understanding the Problem

In this scenario, we have a bag containing 3 green marbles, 11 red marbles, and 6 blue marbles. We are asked to calculate the probability of drawing a blue marble first, followed by a red marble, with replacement.

The first step in solving this problem is to determine the total number of marbles in the bag. This is calculated by adding the quantities of each colored marble:

Total marbles 3 (green) 11 (red) 6 (blue) 20 marbles

Calculating Individual Probabilities

When dealing with independent events (events where the outcome of one event does not influence the outcome of another), the probability of both events occurring is the product of their individual probabilities. Therefore, we start by calculating the probability of drawing a blue marble in the first draw:

Probability (blue first) Number of blue marbles / Total marbles 6/20 0.3

Next, we calculate the probability of drawing a red marble in the second draw. Since we are drawing with replacement, the composition of the bag remains the same:

Probability (red second) Number of red marbles / Total marbles 11/20 0.55

Calculating Combined Probability

To find the combined probability of drawing a blue marble first and a red marble second, we multiply the individual probabilities:

Combined probability (Probability of blue first) * (Probability of red second) (6/20) * (11/20) (66/400) 33/200 0.165

Exploring Different Scenarios

Let's explore a more complex scenario. Suppose we remove two red marbles from the bag. Now, the bag contains 3 green marbles, 9 red marbles (11 - 2), and 6 blue marbles, making a total of 18 marbles. The probabilities would then be:

Probability (blue first) 6/18

Probability (red second) 9/18

Combining these, the probability would be:

Combined probability (6/18) * (9/18) (54/324) 9/54 1/6 or approximately 0.167

Conclusion

The probability of drawing a blue and then a red marble from the original bag with replacement is 0.165. If we modify the bag by removing two red marbles, the probability slightly changes to approximately 0.167. Understanding these calculations is crucial for solving probability problems involving independent events and replacement.

Keywords

By exploring these concepts, we can better understand how to handle more complex probability problems in various scenarios.