Calculate the Probability of Picking a White Ball: A Comprehensive Guide

Probability is a fundamental concept in mathematics, and it finds extensive applications in various fields including statistics, finance, and even real-life decision-making scenarios. One such example is determining the probability of picking a specific colored ball from a bag. Let's dive into the steps required to calculate the probability of picking a white ball from a bag containing 2 white, 3 black, and 4 red balls, when selecting 3 balls at random.

Understanding the Problem

The problem at hand involves a bag containing 9 balls in total: 2 white, 3 black, and 4 red. We want to find out the probability of picking exactly 1 white ball when we randomly select 3 balls from this set. This probability can be calculated using combinatorial mathematics, which deals with the selection of items from a collection without regard to the order of selection.

Step 1: Calculate Total Number of Ways to Pick 3 Balls from 9

The total number of ways to pick 3 balls from 9 without considering the color is given by the combination formula, denoted as (C(n, k) frac{n!}{k!(n - k)!}), where (n) is the total number of items to choose from, (k) is the number of items to be chosen, and (n!) denotes the factorial of (n). For our case: [ n 9 ] [ k 3 ] The formula to calculate the total number of ways is: [ C(9, 3) frac{9!}{3!(9 - 3)!} frac{9!}{3!6!} ] Simplifying the above expression, we get: [ C(9, 3) frac{9 times 8 times 7}{3 times 2 times 1} 84 ] So, there are 84 different ways to pick 3 balls from 9.

Step 2: Calculate Number of Ways to Pick 1 White Ball from the White Balls

Next, we need to calculate the number of ways to pick 1 white ball from the 2 white balls available. This can be done using the combination formula again: [ n 2 ] [ k 1 ] The formula for the number of ways to pick 1 white ball from the 2 white balls is: [ C(2, 1) frac{2!}{1!(2 - 1)!} frac{2!}{1!1!} 2 ] So, there are 2 ways to pick 1 white ball from the 2 white balls.

Step 3: Calculate Number of Ways to Pick 2 Balls from the Remaining 7 Balls

After picking 1 white ball, we need to pick 2 more balls from the remaining 7 balls (3 black and 4 red). Using the combination formula, we get: [ n 7 ] [ k 2 ] The formula for the number of ways to pick 2 balls from the remaining 7 balls is: [ C(7, 2) frac{7!}{2!(7 - 2)!} frac{7!}{2!5!} ] Simplifying the above expression, we get: [ C(7, 2) frac{7 times 6}{2 times 1} 21 ] So, there are 21 ways to pick 2 balls from the remaining 7 balls.

Step 4: Combine the Results to Find the Total Number of Favorable Outcomes

The total number of ways to pick 1 white ball and 2 balls from the remaining 7 is obtained by multiplying the number of ways to pick 1 white ball with the number of ways to pick 2 balls from the remaining 7. [ 2 times 21 42 ] So, there are 42 favorable outcomes.

Step 5: Calculate the Probability

The probability of picking 1 white ball and 2 balls from the remaining 7 balls is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. We already know that the total number of ways to pick 3 balls from 9 is 84. Therefore, the probability is: [ P frac{42}{84} frac{1}{2} times frac{21}{21} frac{1}{2} 0.5 ] Simplifying further, we get: [ P frac{1}{6} 0.1666666… ≈ 16.7% ] So, the probability of picking exactly 1 white ball when 3 balls are picked randomly from the bag is (frac{1}{6}) or approximately 16.7%.

Conclusion

Understanding and applying combinatorial mathematics is crucial for solving probability problems. In this scenario, we calculated the probability of picking a white ball from a bag containing 2 white, 3 black, and 4 red balls when 3 balls are picked at random. By breaking down the problem into smaller steps and using the combination formula, we were able to determine the exact probability. This type of calculation can be useful in various real-world applications such as gaming, statistical analysis, and decision-making.

Frequently Asked Questions

What is the probability of picking 2 white balls?

There are only 2 white balls, so it's not possible to pick 2 white balls out of 3 drawn balls. The probability is 0.

Is there a simpler way to solve this problem?

Using the combination formula is the most straightforward method. However, knowledge of the basics of combinatorics and probability can help simplify the process.

How does this relate to other probability problems?

Understanding principles like combinations and permutations is crucial for solving complex probability problems. These concepts are widely used in fields such as finance, computer science, and data analysis.

Key Takeaways

Combinations are used to calculate the number of ways to choose items from a set without regard to the order. The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes. Understanding and applying combinatorial principles is essential for solving probability problems.