Understanding the Probability of Drawing Two Red Balls Without Replacement

Introduction:

The problem of probability often appears in various scenarios, especially when dealing with events that do not allow for replacement. In this article, we will delve into the specific scenario of drawing two red balls from a bag initially containing 5 red balls, 3 blue balls, and 2 green balls.

The Scenario and Basic Calculations

Let's consider a bag containing a total of 10 balls: 5 red, 3 blue, and 2 green. We want to calculate the probability of drawing two red balls without replacement.

Initial Draw

The probability of drawing the first red ball is simply the number of red balls divided by the total number of balls:

P(first red) 5/10 1/2

Second Draw

Once the first red ball is drawn without replacement, the probability of drawing another red ball changes. The bag now has 4 red balls and 9 total balls. Thus:

P(second red | first red) 4/9

Total Probability Calculation

The combined probability of both events (drawing two red balls in succession) is the product of the individual probabilities:

P(both red) (1/2) * (4/9) 4/18 2/9 ≈ 0.2222

Exploring Other Drawing Combinations

It's worth noting that we can also analyze other scenarios where different colored balls are drawn first, to see how the probability of the second draw changes.

First Blue Ball and Then Red

The probability of first drawing a blue ball and then a red ball:

P(first blue) 3/10

P(second red | first blue) 5/9

Given these probabilities, the combined probability becomes:

P(blue followed by red) (3/10) * (5/9) 15/90 1/6 ≈ 0.1667

First Green Ball and Then Red

Similarly, for the scenario where a green ball is drawn first:

P(first green) 2/10 1/5

P(second red | first green) 5/9

The combined probability is:

P(green followed by red) (1/5) * (5/9) 1/9 ≈ 0.1111

Conclusion

These calculations show the nuanced nature of probability when dealing with events that do not allow for replacement. Each scenario impacts the probability of the next event, making it crucial to consider the changing conditions of the sample space.

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Additional Resources

Keywords: probability of drawing red balls, probability without replacement, combinatorics and probability