Understanding Probability: The Case of Non-Green Balls

Probability theory is an essential tool in statistics and decision-making. This article will explore a specific scenario involving the calculation of the probability of all three balls picked not being green from a bag with a total of 11 balls (5 green, 4 yellow, and 2 blue). We will break down the process step-by-step and provide a clear explanation of the underlying principles.

Scenario Overview

In a bag, there are a total of 11 balls: 5 green, 4 yellow, and 2 blue. The objective is to determine the probability of picking three balls consecutively, without replacement, such that all three balls are not green.

Step-by-Step Calculation

The first step is to calculate the probability of picking a ball that is not green on the first draw. There are 6 balls that are not green (4 yellow 2 blue) out of a total of 11 balls:

P(not green) 6/11

After the first ball is picked and not replaced, there are now 10 balls left in the bag, out of which 5 are not green. Thus, the probability of picking a second ball that is not green is:

P(not green) 5/10 1/2

Consequently, after the second ball is picked and not replaced, there are 9 balls left in the bag, out of which 4 are not green. The probability of picking a third ball that is not green is:

P(not green) 4/9

To find the overall probability, we multiply these individual probabilities together:

P(all non-green) (6/11) * (5/10) * (4/9) 4/33 ≈ 12.12%

Hierarchical Probability Calculation

An alternative method to approach this problem involves calculating the total number of possible outcomes and the number of favorable outcomes. Here is how it works:

The total number of ways to choose any of the balls (5 green 4 yellow 2 blue 11) is the total number of outcomes. The number of ways to choose a ball that is not green (4 yellow 2 blue 6) is the number of favorable outcomes for the first pick:

P(not green) 6/11

After the first pick, the total number of remaining balls is 10, and the number of remaining non-green balls is 5. Therefore, the probability of picking a second non-green ball is:

P(not green) 5/10 1/2

Finally, with 9 balls left and 4 of them being non-green, the probability of picking a third non-green ball is:

P(not green) 4/9

The total probability of all three balls being non-green can be calculated by multiplying the individual probabilities:

P(all non-green) (6/11) * (5/10) * (4/9) 4/33 ≈ 12.12%

Conclusion

Understanding probability is vital for many fields, from statistics to gambling. The case of non-green balls in a bag is a simple yet effective example of how to calculate such probabilities. By breaking down the process into smaller steps, we can accurately determine the overall probability. This method can be generalized to similar scenarios involving no replacement and multiple events.

Keywords

probability, non-green balls, replacement