Calculating the Probability of Picking a Blue Marble

Oct 13, 2025 by miodeya.com 128 views

Introduction

Imagine you have a bag containing 7 marbles of different colors, and you want to predict the number of times you will pick a blue marble if you select one without looking and then put it back 7 times. This problem may seem simple, but it's a great example of how probability and mathematical thinking can help us make informed predictions. In this article, we'll delve into the world of probability and explore the best possible prediction for the number of times you will pick a blue marble.

Understanding Probability

Probability is a measure of the likelihood of an event occurring. In this case, the event is picking a blue marble from the bag. To calculate the probability of an event, we need to know the number of favorable outcomes (picking a blue marble) and the total number of possible outcomes (picking any marble).

Let's assume that there is only one blue marble in the bag, and the remaining 6 marbles are of different colors. The probability of picking a blue marble on the first try is 1/7, since there is only one blue marble out of 7 possible marbles.

The Law of Large Numbers

As we repeat the experiment of picking a marble without looking and then putting it back, the probability of picking a blue marble will remain the same, 1/7, for each individual trial. However, as we perform more trials, the average number of blue marbles picked will approach the expected value, which is the product of the probability of picking a blue marble and the number of trials.

In this case, the expected value is 7 x 1/7 = 1. This means that, on average, we would expect to pick a blue marble once in 7 attempts.

The Binomial Distribution

The binomial distribution is a mathematical model that describes the probability of obtaining k successes (picking a blue marble) in n independent trials (picking a marble without looking and then putting it back), where the probability of success on each trial is p (1/7 in this case).

The binomial distribution is given by the formula:

P(X = k) = (nCk) * p^k * (1-p)^(n-k)

where nCk is the number of combinations of n items taken k at a time, p is the probability of success, and (1-p) is the probability of failure.

Calculating the Probability of Picking a Blue Marble

Using the binomial distribution formula, we can calculate the probability of picking a blue marble k times in 7 attempts. We'll use the following values:

n = 7 (number of trials)
k = 0, 1, 2, 3, 4, 5, 6, or 7 (number of blue marbles picked)
p = 1/7 (probability of picking a blue marble)
(1-p) = 6/7 (probability of not picking a blue marble)

Here are the calculations for each possible value of k:

k	P(X = k)
0	(7C0) * (1/7)^0 * (6/7)^7 ≈ 0.0006
1	(7C1) * (1/7)^1 * (6/7)^6 ≈ 0.0043
2	(7C2) * (1/7)^2 * (6/7)^5 ≈ 0.0204
3	(7C3) * (1/7)^3 * (6/7)^4 ≈ 0.0584
4	(7C4) * (1/7)^4 * (6/7)^3 ≈ 0.1184
5	(7C5) * (1/7)^5 * (6/7)^2 ≈ 0.1804
6	(7C6) * (1/7)^6 * (6/7)^1 ≈ 0.1804
7	(7C7) * (1/7)^7 * (6/7)^0 ≈ 0.0006

The Best Prediction Possible

Based on the calculations above, the probability of picking a blue marble k times in 7 attempts is given by the binomial distribution. However, since we're interested in the best prediction possible, we'll look at the expected value, which is the average number of blue marbles picked.

The expected value is given by:

E(X) = n * p = 7 * 1/7 = 1

This means that, on average, we would expect to pick a blue marble once in 7 attempts.

However, since we're interested in the best prediction possible, we'll look at the most likely outcome, which is the value of k that has the highest probability.

From the calculations above, we can see that the most likely outcome is k = 1, with a probability of approximately 0.0043.

Therefore, the best prediction possible for the number of times you will pick a blue marble is 1 time.

Conclusion

Introduction

In our previous article, we explored the problem of predicting the number of times you will pick a blue marble if you select one without looking and then put it back 7 times. We used the binomial distribution to calculate the probability of picking a blue marble k times in 7 attempts and found that the expected value is 1, which means that, on average, we would expect to pick a blue marble once in 7 attempts. However, the most likely outcome is k = 1, with a probability of approximately 0.0043. In this article, we'll answer some frequently asked questions about the blue marble problem.

Q: What is the probability of picking a blue marble on the first try?

A: The probability of picking a blue marble on the first try is 1/7, since there is only one blue marble out of 7 possible marbles.

Q: What is the expected value of picking a blue marble in 7 attempts?

A: The expected value is given by the product of the probability of picking a blue marble and the number of attempts. In this case, the expected value is 7 x 1/7 = 1.

Q: What is the most likely outcome of picking a blue marble in 7 attempts?

A: The most likely outcome is k = 1, with a probability of approximately 0.0043.

Q: Can I use the binomial distribution to calculate the probability of picking a blue marble in more than 7 attempts?

A: Yes, you can use the binomial distribution to calculate the probability of picking a blue marble in more than 7 attempts. However, the calculations become more complex and may require the use of a calculator or computer software.

Q: What if there are multiple blue marbles in the bag?

A: If there are multiple blue marbles in the bag, the probability of picking a blue marble on the first try is greater than 1/7. For example, if there are 2 blue marbles out of 7 possible marbles, the probability of picking a blue marble on the first try is 2/7.

Q: Can I use the blue marble problem to model real-world situations?

A: Yes, the blue marble problem can be used to model real-world situations where there are multiple outcomes and a fixed probability of success. For example, you can use the blue marble problem to model the probability of a coin landing heads up or tails up in a series of flips.

Q: What is the relationship between the blue marble problem and the law of large numbers?

A: The blue marble problem is an example of the law of large numbers, which states that the average of a large number of independent and identically distributed random variables will converge to the expected value. In this case, the expected value is 1, which means that, on average, we would expect to pick a blue marble once in 7 attempts.

Q: Can I use the blue marble problem to teach probability and statistics to students?

A: Yes, the blue marble problem is a great way to teach probability and statistics to students. It's a simple and intuitive example that can be used to illustrate key concepts such as probability, expected value, and the law of large numbers.

Conclusion

In this article, we answered some frequently asked questions about the blue marble problem. We hope that this Q&A session has been helpful in clarifying any doubts you may have had about the problem. If you have any further questions, please don't hesitate to ask.