Calculating Probability of Drawing a Green Ball

In this article, we will explore the probability of drawing a green ball from a bag containing various colored balls. We'll go through the steps necessary to solve these kinds of problems, including the application of basic probability principles and the law of total probability.

Introduction to Probability in Bag of Balls

Consider a bag that contains a total of 5 red, 3 green, and 2 blue balls. The total number of balls in the bag can be calculated as:

5 (Red) 3 (Green) 2 (Blue) 10 balls

If we were to randomly draw a ball from the bag without replacement, what is the probability that we draw a green ball?

Step 1: Basic Probability Calculation

To calculate the probability of drawing a green ball in one single draw, we use the formula:

Probability Number of favorable outcomes / Total number of possible outcomes

The number of favorable outcomes (drawing a green ball) is 3, and the total number of possible outcomes (total balls) is 10.

Therefore, the probability of drawing a green ball in one single draw is:

P(green) 3 / 10 0.3

Step 2: Calculating the Probability of Drawing a Green Ball as the Second Ball

Let's consider a different scenario where we are interested in the probability that the second ball drawn is green. To solve this problem, we can use the law of total probability. We will consider different scenarios for the first ball drawn and calculate the probability for each case.

Scenario 1: First Ball is Red

Probability of drawing a red ball first: PR 5 / 10 1 / 2 Remaining balls after drawing a red ball: 4 red, 3 green, and 2 blue (total 9 balls) Probability that the second ball is green: PG R 3 / 9 1 / 3 Combined probability for this scenario: (1 / 2) * (1 / 3) 1 / 6

Scenario 2: First Ball is Blue

Probability of drawing a blue ball first: PB 3 / 10 3 / 10 Remaining balls after drawing a blue ball: 5 red, 2 green, and 1 blue (total 8 balls) Probability that the second ball is green: PG B 2 / 8 1 / 4 Combined probability for this scenario: (3 / 10) * (1 / 4) 3 / 40

Scenario 3: First Ball is Green

Probability of drawing a green ball first: PG 3 / 10 3 / 10 Remaining balls after drawing a green ball: 5 red, 2 green, and 2 blue (total 9 balls) Probability that the second ball is green: PG G 2 / 9 Combined probability for this scenario: (3 / 10) * (2 / 9) 1 / 15

Final Step: Summing Up All Probabilities

To find the overall probability that the second ball drawn is green, we sum up the probabilities of each individual scenario:

PG (1 / 2) * (1 / 3) (3 / 10) * (1 / 4) (3 / 10) * (2 / 9)

Simplifying the fractions:

(1 / 2) * (1 / 3) 1 / 6 (3 / 10) * (1 / 4) 3 / 40 (3 / 10) * (2 / 9) 6 / 90 1 / 15

Converting them to a common denominator (60), we get:

1 / 6 10 / 60 3 / 40 9 / 60 1 / 15 4 / 60

Adding these together:

PG 10 / 60 9 / 60 4 / 60 23 / 60

Thus, the probability that the second ball drawn is green is:

PG 23 / 60

So, the probability that the second ball drawn is green is approximately 23 / 60 or about 0.3833.

By following these detailed steps, we can accurately calculate the probability of drawing a specific colored ball under various scenarios, using both basic probability and more advanced techniques like the law of total probability.