Probability of Picking a Specific Color Marble When Your Friend Chooses One

Imagine you have a bag containing three marbles: two blue, one red, and one black. You're considering picking out a marble at random, but your friend decides to pick one marble as well. Your friend's selection certainly influences your likelihood of picking a specific color. In this article, we will delve into the concept of probability and explore the possible outcomes.

Understanding the Scenario

You have a bag with three different colored marbles: two blue, one red, and one black. Your friend decides to pick one marble, and it's important to note that your friend's choice is not up to you. Given that your friend has chosen one marble, you are left with either two or one of each color, depending on the color of the marble your friend has chosen.

Calculating the Probability

Let's break it down step by step. First, consider the initial probability of picking a specific color from the bag of three marbles:

Probability of picking a blue marble: 2/3 Probability of picking a red marble: 1/3 Probability of picking a black marble: 1/3

Now, when your friend picks one marble, you are left with two marbles in the bag. Let's consider each scenario:

Scenario 1: Your Friend Picks a Blue Marble

If your friend picks a blue marble, you now have two marbles left: one blue and one red, and one black. The probability of picking a blue marble from the remaining two marbles is:

Probability of picking a blue marble: 1/2 Probability of picking a red marble: 1/2 Probability of picking a black marble: 0

Scenario 2: Your Friend Picks a Red Marble

If your friend picks a red marble, you now have two marbles left: two blue, and one black. The probability of picking a blue marble from the remaining two marbles is:

Probability of picking a blue marble: 2/2 1 Probability of picking a red marble: 0 Probability of picking a black marble: 0

Scenario 3: Your Friend Picks a Black Marble

If your friend picks a black marble, you now have two marbles left: two blue, and one red. The probability of picking a blue marble from the remaining two marbles is:

Probability of picking a blue marble: 2/2 1 Probability of picking a red marble: 0 Probability of picking a black marble: 0

Combining Probabilities

To find the overall probability of picking a specific color marble, we need to combine the probabilities of each scenario with the probability of your friend picking that color:

Probability that your friend picks a red marble and you pick a blue marble: (1/3) * (1/2) 1/6 Probability that your friend picks a red marble and you pick a red marble: (1/3) * (1/2) 1/6 Probability that your friend picks a black marble and you pick a blue marble: (1/3) * 1 1/3 Probability that your friend picks a blue marble and you pick a blue marble: (2/3) * (1) 2/3

Conclusion

Therefore, the overall probability of picking a specific color marble when your friend's choice is taken into account is:

Probability of picking a blue marble: 2/3 1/6 5/6 Probability of picking a red marble: 1/6 Probability of picking a black marble: 1/3

In this scenario, the probability is not zero, as your friend's choice influences the outcome. Understanding these probabilities can help you make more informed decisions and predictions in various real-life situations.