Independent Events Probability P(A) = 0.70 And P(B) = 0.20

Introduction to Independent Events

In the realm of probability theory, understanding the concept of independent events is crucial for solving a wide array of problems. Two events are considered independent if the occurrence of one event does not affect the probability of the other event occurring. This fundamental principle has significant applications in various fields, including statistics, finance, and engineering. This article delves into the conditions under which events A and B, with probabilities P(A) = 0.70 and P(B) = 0.20, are deemed independent. We will explore different scenarios and use the principles of probability to determine the correct condition for independence. To grasp the essence of independent events, it's essential to first define some key terms and formulas. The probability of an event A, denoted as P(A), is a measure of the likelihood that event A will occur. When we talk about two events A and B, we often consider their joint occurrence, represented by P(A and B), which is the probability that both A and B happen simultaneously. Additionally, we look at the probability of either A or B occurring, denoted as P(A or B), which includes the cases where A happens, B happens, or both happen. These probabilities are interconnected through specific formulas that govern their relationships. The formula for the probability of the union of two events is given by: P(A or B) = P(A) + P(B) - P(A and B). This formula is crucial because it accounts for the overlap between the events; the term P(A and B) is subtracted to avoid double-counting the cases where both events occur. Now, let's focus on what it means for two events to be independent. The mathematical condition for independence is elegantly simple: two events A and B are independent if and only if P(A and B) = P(A) * P(B). This equation states that the probability of both events occurring is simply the product of their individual probabilities. This makes intuitive sense because if the events are independent, the occurrence of one should not change the likelihood of the other, and their probabilities should multiply. Understanding this condition is paramount for analyzing the given scenarios and determining when events A and B, with probabilities 0.70 and 0.20 respectively, are truly independent. This article will dissect the various options provided, applying this principle to each to identify the correct answer. We will also explore why the other options are incorrect, further solidifying our understanding of independent events. By the end of this discussion, you will have a clear and comprehensive grasp of how to assess independence and its implications in probability.

Analyzing the Given Probabilities: P(A) = 0.70 and P(B) = 0.20

Given the probabilities P(A) = 0.70 and P(B) = 0.20, the task is to determine the condition under which events A and B are independent. As established, the critical condition for independence is P(A and B) = P(A) * P(B). We can calculate the product of the individual probabilities: P(A) * P(B) = 0.70 * 0.20 = 0.14. This calculation provides a benchmark for assessing the options. For events A and B to be independent, the probability of their joint occurrence, P(A and B), must equal 0.14. This is a direct application of the definition of independence and provides a concrete value to compare against. Now, let's examine each of the provided options in light of this calculation. Option A states P(A or B) = 0.90. To evaluate this, we recall the formula for the union of two events: P(A or B) = P(A) + P(B) - P(A and B). If P(A or B) = 0.90, we can rearrange this formula to solve for P(A and B): P(A and B) = P(A) + P(B) - P(A or B). Substituting the given values, we get P(A and B) = 0.70 + 0.20 - 0.90 = 0. This result indicates that if P(A or B) were 0.90, P(A and B) would be 0, which does not satisfy the condition for independence (P(A and B) = 0.14). Thus, option A is not the correct condition for independence. Option B states P(A or B) = 0.14. Using the same formula for the union of two events, we can solve for P(A and B): P(A and B) = P(A) + P(B) - P(A or B). Substituting the given values, we get P(A and B) = 0.70 + 0.20 - 0.14 = 0.76. This result shows that if P(A or B) were 0.14, P(A and B) would be 0.76, which also does not match the required value of 0.14 for independence. Thus, option B is also not the correct condition. Option C states P(A and B) = 0.14. This option directly matches our calculated value for the product of the individual probabilities, P(A) * P(B) = 0.14. According to the definition of independent events, this condition perfectly satisfies the requirement for independence. Therefore, option C is the correct condition for events A and B to be independent. Option D is incomplete, and we cannot evaluate it without additional information. However, based on our analysis, option C is the only one that aligns with the mathematical criterion for independence. This detailed analysis of each option not only identifies the correct answer but also provides a thorough understanding of why the other options do not satisfy the conditions for independence. This approach strengthens the comprehension of the concepts involved and allows for a more nuanced understanding of probability theory.

Detailed Examination of the Options and the Independence Condition

To further solidify the understanding of independent events, it is crucial to dissect each option and demonstrate why only one satisfies the condition P(A and B) = P(A) * P(B). We have already established that P(A) = 0.70 and P(B) = 0.20, and their product, which must equal P(A and B) for independence, is 0.14. Let's revisit each option with a more in-depth analysis. Option A proposes P(A or B) = 0.90. As previously calculated, using the formula P(A or B) = P(A) + P(B) - P(A and B), we can find P(A and B): P(A and B) = P(A) + P(B) - P(A or B) = 0.70 + 0.20 - 0.90 = 0. This result implies that events A and B are mutually exclusive, meaning they cannot occur simultaneously. If A and B were independent with these probabilities, P(A and B) should be 0.14, not 0. Thus, this option contradicts the independence condition. The implications of P(A or B) = 0.90 are significant. It suggests a strong likelihood that either A or B will occur, but their simultaneous occurrence is impossible. This scenario is distinct from independence, where the events can occur together, and their joint probability is the product of their individual probabilities. Understanding this difference is vital for accurate probabilistic reasoning. Option B suggests P(A or B) = 0.14. Again, using the formula P(A or B) = P(A) + P(B) - P(A and B), we can solve for P(A and B): P(A and B) = P(A) + P(B) - P(A or B) = 0.70 + 0.20 - 0.14 = 0.76. In this case, P(A and B) is significantly higher than the expected value of 0.14 for independent events. This discrepancy clearly indicates that events A and B are not independent under this condition. The value of P(A and B) = 0.76 suggests a strong positive association between the events. This means that if A occurs, B is also highly likely to occur, and vice versa. This type of dependency is the opposite of independence, where the occurrence of one event does not influence the probability of the other. Option C states P(A and B) = 0.14. This directly satisfies the independence condition P(A and B) = P(A) * P(B) = 0.70 * 0.20 = 0.14. When the joint probability of A and B equals the product of their individual probabilities, the events are, by definition, independent. This option is the only one that aligns perfectly with the mathematical criterion for independence. Option D is incomplete, making it impossible to assess its validity. Without a specific probability value or condition, we cannot determine whether it supports or contradicts the independence of events A and B. However, given that option C satisfies the independence condition, it is the most accurate answer. This thorough examination of each option underscores the importance of understanding the mathematical definition of independence. By applying the formula P(A and B) = P(A) * P(B) and comparing the calculated joint probability with the given conditions, we can definitively determine whether events are independent. This analytical approach is essential for solving probability problems and making informed decisions based on probabilistic data.

The Correct Condition for Independence: P(A and B) = 0.14

In summary, the condition for events A and B to be independent when P(A) = 0.70 and P(B) = 0.20 is P(A and B) = 0.14. This conclusion is derived directly from the fundamental definition of independence in probability theory. Two events are independent if and only if the probability of both events occurring simultaneously is equal to the product of their individual probabilities. This can be mathematically expressed as P(A and B) = P(A) * P(B). Applying this principle to the given probabilities, we find that P(A) * P(B) = 0.70 * 0.20 = 0.14. Therefore, if P(A and B) is indeed 0.14, events A and B are independent. To reiterate, the independence of events means that the occurrence of one event does not affect the probability of the other event occurring. This is a crucial concept in various fields, including statistics, finance, and machine learning. Understanding and correctly applying the definition of independence is essential for accurate probabilistic reasoning and decision-making. The incorrect options provided, such as P(A or B) = 0.90 and P(A or B) = 0.14, highlight the importance of distinguishing between different probability relationships. The probability of the union of two events, P(A or B), involves a different calculation that considers the possibility of both events occurring, as reflected in the formula P(A or B) = P(A) + P(B) - P(A and B). These options lead to values of P(A and B) that do not align with the independence condition, thus demonstrating that the joint occurrence of events is influenced by factors beyond their individual probabilities. The meticulous analysis of each option underscores the need for a precise understanding of probabilistic relationships. By adhering to the definition of independence and applying the relevant formulas, we can accurately determine whether events are independent. This knowledge is foundational for solving more complex probability problems and for making informed judgments in real-world scenarios where probabilistic assessments are necessary. In conclusion, the correct condition for independence is unequivocally P(A and B) = 0.14, as it is the only condition that directly satisfies the mathematical definition of independence for the given probabilities P(A) = 0.70 and P(B) = 0.20. This understanding forms a cornerstone for further exploration of probability and its applications.