Probability Calculation: Picking Two Black Balls Without Replacement

Probability is a fundamental concept in mathematics and statistics, often used in real-world scenarios such as games, experiments, and various decision-making processes. One such problem involves a bag that contains 10 black and 5 yellow balls. We will explore the scenario where two balls are picked at random, one after the other, without replacement, and calculate the probability that both balls are black.

Problem Statement

The problem can be framed as follows: What is the probability of picking two balls from a bag containing 10 black and 5 yellow balls, with the condition that the second ball is picked without replacement?

Step-by-Step Solution

Let's break this down step by step, using the given information and basic principles of probability.

Step 1: Total Number of Balls

First, we note the total number of balls in the bag.

Total number of balls 10 (black) 5 (yellow) 15 balls

Step 2: Probability of the First Ball Being Black

The probability of the first ball being black is the number of black balls divided by the total number of balls.

Probability (first ball is black) 10/15 2/3

Step 3: Adjusting for the Second Draw

After the first ball is picked and not replaced, the total number of balls decreases by one, and the number of black balls also decreases by one if the first ball was black.

Total number of balls after first draw 14

Number of black balls left 10 - 1 9

Step 4: Probability of the Second Ball Being Black

The probability of the second ball being black is now the number of remaining black balls divided by the updated total number of balls.

Probability (second ball is black) 9/14

Step 5: Combined Probability

Since the events are dependent, we multiply the probabilities of each event. Therefore, the combined probability of both events happening is:

Probability (both balls are black) (2/3) * (9/14) 18/42 3/7

This shows that the probability that both balls picked are black is 3/7.

Conclusion

In summary, the probability of picking two black balls from a bag containing 10 black and 5 yellow balls, without replacement, is 3/7. This result can be verified by the step-by-step calculation provided above. Such problems help to understand and apply probability concepts in real-world scenarios.

Understanding and solving problems like these can enhance one's analytical and logical reasoning skills, which are invaluable in various fields, including business, engineering, and data analysis.

Further Reading

For a deeper understanding of probability, you may explore related topics such as:

Conditional Probability Expected Value Binomial Distribution

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