Probability Of Picking At Least One Black Ball

In the realm of probability, understanding how to calculate the likelihood of specific events is crucial. This article delves into a classic probability problem involving a bag containing balls of different colors. We will explore the scenario where two balls are drawn from the bag without replacement and calculate the probability of at least one ball being black. This problem is a great example of how to apply fundamental probability concepts such as combinations and complementary events. This exploration will not only enhance your understanding of probability calculations but also demonstrate how these principles can be applied to real-world scenarios. The core of the problem lies in identifying all possible outcomes and then determining the outcomes that satisfy the given condition – in this case, picking at least one black ball. We will dissect this problem step-by-step, providing a clear and concise methodology for solving it. The understanding of combinations, permutations, and the concept of complementary probability will be instrumental in navigating this problem successfully. By the end of this article, you will be equipped with the knowledge and skills to tackle similar probability problems with confidence.

A bag contains 4 black balls and 5 red balls, all similar in shape and size. If two balls are picked from the bag without replacement, what is the probability that at least one ball is black?

This problem introduces us to the concept of probability without replacement, a key aspect of many real-world scenarios. The phrase "at least one ball is black" requires careful consideration as it encompasses multiple scenarios: one black ball and one red ball, or two black balls. To solve this problem efficiently, we will explore different approaches, including calculating the probability of the complementary event (i.e., no black balls are picked) and subtracting it from 1. Understanding the nuances of this problem will not only help in solving it but also in grasping the broader principles of probability calculations in scenarios involving dependent events. Dependent events, in this context, are events where the outcome of one affects the outcome of the other, as is the case when balls are drawn without replacement. Therefore, the problem serves as a gateway to understanding more complex probability situations where outcomes are not independent.

Before diving into the solution, let's recap some essential probability concepts:

• Probability: The likelihood of an event occurring, expressed as a ratio of favorable outcomes to the total number of possible outcomes.
• Without Replacement: Once an item is selected, it is not returned to the pool, affecting the probabilities of subsequent selections.
• At Least One: This implies one or more, covering multiple scenarios.
• Combinations: A way of selecting items from a collection, such that the order of selection does not matter.

Probability, at its core, is about quantifying uncertainty. It allows us to make informed decisions and predictions in situations where outcomes are not deterministic. The concept of "without replacement" introduces a dependency between events, meaning that the probability of the second event depends on the outcome of the first. This is in contrast to scenarios where items are replaced, making the events independent. The phrase "at least one" is crucial as it often requires considering multiple scenarios or using the concept of complementary probability, which simplifies calculations. Understanding combinations is vital as it helps us count the number of ways to select items without regard to order, a common requirement in probability problems. These basic concepts are the building blocks for tackling more complex probability problems and are essential for anyone seeking to master this area of mathematics.

There are a couple of ways to solve this problem:

1. Direct Method: Calculate the probabilities of the two favorable scenarios (one black and one red, or two black) and add them up.
2. Complementary Method: Calculate the probability of the complementary event (no black balls are picked) and subtract it from 1.

Both methods are valid, but the complementary method is often more straightforward in problems involving "at least one." Let's explore both approaches in detail to illustrate their application and demonstrate why the complementary method might be preferred in this case. The direct method involves breaking down the problem into distinct cases and calculating the probability of each case separately. This can be more time-consuming and error-prone, especially when the number of cases increases. The complementary method, on the other hand, focuses on the probability of the event not happening, which can sometimes be simpler to calculate. By subtracting this probability from 1, we obtain the probability of the event we are interested in. This approach leverages the fundamental principle that the sum of the probabilities of an event and its complement is always 1. Understanding both methods provides flexibility in problem-solving and allows one to choose the most efficient approach based on the specific problem at hand.

Method 1: Direct Method

Scenario 1: One Black and One Red Ball

First, we calculate the probability of picking one black ball and one red ball. This can happen in two ways: picking a black ball first and then a red ball, or picking a red ball first and then a black ball.

• Probability of Black then Red: (4/9) * (5/8) = 20/72
• Probability of Red then Black: (5/9) * (4/8) = 20/72

Adding these probabilities together gives us the total probability of picking one black and one red ball: 20/72 + 20/72 = 40/72.

This scenario highlights the importance of considering all possible sequences when calculating probabilities in dependent events. The probability of picking a black ball first is 4/9, as there are 4 black balls out of a total of 9. After picking a black ball, there are 8 balls left, 5 of which are red, so the probability of picking a red ball next is 5/8. Similarly, we calculate the probability of picking a red ball first and then a black ball. The crucial step is to recognize that either sequence satisfies the condition of having one black and one red ball, so we add the probabilities of the two sequences to get the total probability for this scenario. This approach underscores the importance of careful enumeration and calculation when dealing with probabilities involving multiple events and sequences.

Scenario 2: Two Black Balls

Next, we calculate the probability of picking two black balls. The probability of picking the first black ball is 4/9. After picking one black ball, there are 3 black balls and 8 total balls remaining.

• Probability of Two Black Balls: (4/9) * (3/8) = 12/72

This calculation demonstrates how the probability changes after the first ball is drawn, emphasizing the concept of "without replacement." The initial probability of picking a black ball is 4/9. Once a black ball is removed, the composition of the bag changes, and the probability of picking another black ball becomes 3/8. Multiplying these probabilities gives us the overall probability of picking two black balls in succession. This straightforward calculation highlights the sequential nature of dependent events and how the outcome of one event directly influences the probabilities of subsequent events. It's a clear illustration of how probabilities must be adjusted to reflect the changing state of the system as balls are removed from the bag without being replaced.

Total Probability (Direct Method)

To find the total probability of at least one black ball, we add the probabilities from the two scenarios:

Total Probability = Probability (One Black and One Red) + Probability (Two Black Balls)

Total Probability = 40/72 + 12/72 = 52/72

Simplifying the fraction, we get 52/72 = 13/18.

This final step combines the probabilities of the mutually exclusive scenarios that satisfy the condition of "at least one black ball." By adding the probabilities of picking one black and one red ball and picking two black balls, we arrive at the overall probability of the event occurring. The simplification of the fraction to 13/18 provides the probability in its simplest form, making it easier to interpret and compare with other probabilities. This summation underscores the additive nature of probabilities for mutually exclusive events, a fundamental principle in probability theory. The final result gives a clear quantitative measure of the likelihood of the event "at least one black ball" occurring when two balls are drawn from the bag.

Method 2: Complementary Method

Probability of No Black Balls

The complementary event to "at least one black ball" is "no black balls," which means we pick two red balls. The probability of picking a red ball first is 5/9. After picking a red ball, there are 4 red balls and 8 total balls remaining.

• Probability of Two Red Balls: (5/9) * (4/8) = 20/72

This calculation represents a crucial step in the complementary method, where we focus on the probability of the event not happening. By calculating the probability of picking two red balls, we are essentially finding the probability of the complementary event to "at least one black ball." The initial probability of picking a red ball is 5/9, reflecting the number of red balls relative to the total number of balls. After one red ball is removed, the probabilities shift, and the probability of picking another red ball becomes 4/8. Multiplying these probabilities gives us the overall probability of the event "two red balls" occurring. This calculation illustrates the power of the complementary method in simplifying probability problems, especially those involving "at least one" conditions.

Total Probability (Complementary Method)

The probability of at least one black ball is 1 minus the probability of no black balls:

Total Probability = 1 - Probability (No Black Balls)

Total Probability = 1 - 20/72 = 52/72

Simplifying the fraction, we get 52/72 = 13/18.

This final step elegantly demonstrates the principle of complementary probability. By subtracting the probability of the complementary event (no black balls) from 1, we directly obtain the probability of the event we are interested in (at least one black ball). The result, 52/72, simplifies to 13/18, matching the result obtained using the direct method. This consistency reinforces the validity of both approaches and highlights the flexibility in choosing the most efficient method for solving a particular probability problem. The complementary method is particularly useful when calculating the probability of "at least one" scenarios, as it often involves fewer calculations than directly considering all possible cases. This approach underscores the importance of strategic problem-solving in probability and the power of leveraging complementary events to simplify complex calculations.

Both methods yield the same result: the probability of picking at least one black ball is 13/18. The complementary method provides a more efficient solution in this case. This problem demonstrates the importance of understanding fundamental probability concepts and choosing the right approach for solving probability problems. Whether using the direct method or the complementary method, a solid understanding of the underlying principles is crucial for accurate calculations. The complementary method, in particular, showcases how focusing on the opposite of the desired event can sometimes lead to a simpler and more elegant solution. Ultimately, proficiency in probability requires a combination of conceptual understanding, computational skills, and strategic problem-solving. By mastering these elements, one can confidently tackle a wide range of probability problems and apply these principles to real-world scenarios.

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