Introduction

In this article, we'll explore the concept of probability in the context of drawing balls from a bag. Specifically, we'll calculate the probability of drawing exactly one green ball when two balls are drawn at random from a bag containing 3 red, 4 green, and 2 blue balls. We'll also discuss the general approach to calculating such probabilities using combinations and the combination formula.

Calculating the Probability of Drawing Exactly One Green Ball

Step 1: Determine the Total Number of Balls

The bag contains 3 red, 4 green, and 2 blue balls. The total number of balls in the bag is the sum of these quantities:

3 (red) 4 (green) 2 (blue) 9 balls

Step 2: Calculate the Total Number of Ways to Choose 2 Balls From 9

When choosing 2 balls from 9, we use the combination formula, denoted as Cnk:

C92 9! {2!(9-2)!}

This simplifies to:

9 × 8 {2 × 1} 36

Step 3: Calculate the Number of Favorable Outcomes for Choosing Exactly One Green Ball

To have exactly one green ball, we can choose:

1 green ball from the 4 available green balls 1 non-green ball from the remaining 5 balls (3 red 2 blue)

The number of ways to choose 1 green ball from 4 is:

C41 4

The number of ways to choose 1 non-green ball from 5 is:

C51 5

The total number of favorable outcomes is the product of these two combinations:

4 × 5 20

Step 4: Calculate the Probability

The probability of drawing exactly one green ball is the ratio of favorable outcomes to total outcomes:

P(Exactly one green) 20 {36} 5 {9}

Additional Examples and Scenarios

Simultaneously Drawing Two Balls at Once

In another scenario, if two balls are drawn at once, the total number of possibilities is calculated as:

(3 red 4 green 2 blue) × (3 red 4 green 2 blue) 9 × 9 81

The number of ways to draw 1 red and 1 green ball, 1 blue and 1 green ball, and 2 green balls are:

R × G 3 × 4 12 B × G 2 × 4 8 G × G 4 × 3 12

The probability of drawing a green ball from the bag is given by:

P(Green) (12 8 12) {81} 32 {81} ≈ 0.395

General Probability Calculation

To find the probability of drawing a green ball from a bag with 4 green balls and 12 total balls (3 red 4 green 5 blue), the calculation is as follows:

P(Green) 4 {12} 1 {3}

This simplifies to:

P(Green) 0.333

Conclusion

Understanding the principles of combinations and the combination formula is crucial in calculating probabilities in scenarios like drawing balls from a bag. By breaking down the problem into manageable steps, we can accurately determine the likelihood of specific outcomes.