Probability of Drawing a Blue Disc from Randomly Chosen Bags

Imagine two bags, each containing a different number and color of discs. Bag A contains four red and three blue discs, while Bag B has two red and five blue discs. A bag is chosen at random, and a disc is then drawn from the chosen bag. This article explores the probability of drawing a blue disc from the randomly chosen bag and delves into the mathematical reasoning behind it.

Probability of Drawing a Blue Disc from a Single Experiment

To determine the probability of drawing a blue disc from either bag, we need to consider the probability of choosing each bag and the probability of drawing a blue disc from each bag.

Bag A

Bag A contains 4 red and 3 blue discs, a total of 7 discs. The probability of drawing a blue disc from Bag A is calculated as:

P(Blue from Bag A) Number of blue discs / Total number of discs 3/7

Bag B

Bag B contains 2 red and 5 blue discs, a total of 7 discs. The probability of drawing a blue disc from Bag B is calculated as:

P(Blue from Bag B) Number of blue discs / Total number of discs 5/7

Combined Probability

Since either Bag A or Bag B is chosen at random, the probability of choosing either bag is 1/2. Therefore, the combined probability of drawing a blue disc from a randomly chosen bag is calculated as:

P(Blue)  P(Choosing Bag A) * P(Blue from Bag A)   P(Choosing Bag B) * P(Blue from Bag B)           (1/2) * (3/7)   (1/2) * (5/7)           (3/14)   (5/14)           8/14           4/7          ≈ 57.14%

Repeating the Experiment Multiple Times

Let's consider an experiment where the described scenario is repeated 7 times. Each time, a bag is chosen and a disc is drawn. We will then determine the average outcome if this experiment is repeated many times.

Bag Filling Experiment

If we repeat the experiment 7 times and then repeat the same setup many times, we can observe that each bag is chosen approximately the same number of times. Therefore, the average composition of the bags can be approximated.

For the second experiment, the probability of drawing a blue disc from Bag B (when Bag B is chosen) is 5/9, and if Bag A is chosen, the probability of drawing a blue disc is 6/9. To find the weighted average, we calculate:

Weighted Average (P(Blue))  (4 * 5/9   3 * 6/9) / 7                             (20/9   18/9) / 7                             (38/9) / 7                             38/63                            ≈ 0.60317

This result indicates that the probability of drawing a blue disc from the bags, on average, is approximately 60.32%.

Conclusion

To summarize, the probability of drawing a blue disc from a randomly chosen bag (Bag A or Bag B) is approximately 57.14%, while the weighted average probability, over many repetitions of the experiment, is approximately 60.32%. These calculations demonstrate the importance of considering all possible outcomes and their probabilities when dealing with random selection scenarios.