Probability of Marbles Drawn from a Jar

In this article, we will explore the probabilities of drawing marbles from a jar with and without replacement. Specifically, we will calculate the probability of drawing two marbles, one by one, from a jar containing 4 blue marbles and 7 yellow marbles.

Understanding the Problem

The problem involves a jar containing 4 blue marbles and 7 yellow marbles. We are to draw two marbles one by one with replacement, meaning the first marble is put back into the jar before the second draw. We will calculate two specific probabilities:

Part (a): The probability that both marbles drawn are yellow.

Part (b): The probability that the second marble drawn is blue.

Calculating the Probabilities

Part (a): Probability that Both Marbles Are Yellow

To calculate this, we first determine the total number of marbles in the jar:

Total Marbles 4 (blue) 7 (yellow) 11 marbles

Since we are drawing with replacement, the probabilities remain constant for each draw.

The probability of drawing a yellow marble on the first draw is:

P(Yellow on first draw) 7/11

Similarly, the probability of drawing a yellow marble on the second draw is:

P(Yellow on second draw) 7/11

The combined probability of both draws being yellow is the product of the individual probabilities:

P(Both Yellow) P(Yellow on first draw) * P(Yellow on second draw) (7/11) * (7/11) 49/121

Part (b): Probability that the Second Marble is Blue

In this part, we need to consider the scenarios for the first draw because we are interested in the second marble being blue. However, the probability of the second marble being blue is independent of the outcome of the first draw since we are drawing with replacement.

The probability of drawing a blue marble on the second draw is:

P(Blue on second draw) 4/11

The first marble can be either a blue or a yellow, but the probability of the second marble being blue is still (4/11) regardless of the color of the first marble. Therefore, the probability that the second marble is blue is:

P(Second Blue) 4/11

Conclusion

Based on our calculations, we have:

The probability that both marbles are yellow is 49/121. The probability that the second marble is blue is 4/11.

These calculations demonstrate the application of basic probability concepts, including the independence of events and the use of replacement to maintain constant probabilities.

Related Keywords

probability of drawing marbles replacement probability independent events in probability

Frequently Asked Questions (FAQs)

Q: What is the probability of drawing two yellow marbles with replacement?

A: The probability of drawing two yellow marbles with replacement is 49/121, as calculated.

Q: Does the probability of drawing a blue marble change if the first marble drawn is yellow?

A: No, because we are drawing with replacement, the probability of drawing a blue marble remains constant at 4/11 for the second draw, regardless of the first marble's color.

We hope this article helps you understand the concept of probability in the context of drawing marbles from a jar. If you have any other questions or need further clarification, feel free to reach out!