Understanding the Probability of Picking Two Marbles of Different Colors

When dealing with probability in a scenario involving marbles or any other identical objects, it's essential to break down the problem into manageable steps. This article will walk you through the process of calculating the probability of picking two marbles of different colors from a box containing 2 white and 3 blue marbles, with the condition that the marbles are picked one after the other without replacement.

The Scenario and Problem Statement

In this problem, we have a box containing 2 white and 3 blue identical marbles. The task is to calculate the probability of picking two marbles of different colors, one after the other, without replacing the first marble. This type of problem is often encountered in probability theory and can be solved using a step-by-step approach.

Step-by-Step Calculation

Step 1: Calculate the Total Ways to Pick 2 Marbles

First, we need to determine the total number of ways to choose 2 marbles from the 5 available marbles. This can be done using the combination formula:

Total ways (binom{5}{2}) (frac{5!}{2!(5-2)!}) (frac{5 times 4}{2 times 1}) 10

Step 2: Calculate the Favorable Outcomes

We need to find the number of ways to pick one white and one blue marble. We will do this by calculating the number of ways to pick 1 white marble and 1 blue marble separately and then multiplying the results.

Choosing 1 White Marble: There are 2 white marbles, so the number of ways to choose 1 white marble is:

(binom{2}{1}) 2

Choosing 1 Blue Marble: There are 3 blue marbles, so the number of ways to choose 1 blue marble is:

(binom{3}{1}) 3

Total Favorable Outcomes: The total number of ways to choose 1 white and 1 blue marble is:

2 (times) 3 6

Step 3: Calculate the Probability

Now, we can calculate the probability of picking two marbles of different colors:

Probability (frac{text{Number of favorable outcomes}}{text{Total ways}}) (frac{6}{10}) (frac{3}{5})

Conclusion

Thus, the probability of picking two marbles of different colors is (frac{3}{5}) or 0.60. This means there is a 60% chance of picking a white and a blue marble in succession without replacement.

Additional Scenarios

The calculations above can be extended to similar problems involving other numbers of marbles. For example, if you had 2 white and 2 blue marbles, the probability for picking two different colors would be (frac{4}{4} times frac{2}{3} (frac{2}{3}) or approximately 0.67. However, if you are picking two marbles with replacement, the probability would be (frac{4}{4} times frac{4}{4} (frac{4}{4}) 1, reflecting the higher likelihood of different colors due to the resetting of the conditions after each pick.

Key Takeaways

1. **Understanding the Total Outcomes:** The combination formula helps in calculating the total number of ways to pick the marbles.

2. **Favorable Outcomes:** Separate calculations for each type of favorable outcomes (in this case, one white and one blue) are necessary.

3. **Probability Calculation:** Dividing the number of favorable outcomes by the total number of outcomes gives the desired probability.

Further Exploration

Exploring similar problems involving different numbers of marbles or different colored marbles can help solidify your understanding of probability in various scenarios. Remember, the key is to carefully break down the problem and apply the appropriate mathematical principles.