Finding Missing Probability Step-by-Step Solution

Hey guys! Today, we're diving into the fascinating world of probability to solve a common yet crucial problem: finding a missing probability. Probability, at its core, is the measure of the likelihood that an event will occur. It's a concept we encounter daily, from weather forecasts predicting rain to the odds of winning the lottery. Understanding how to calculate and manipulate probabilities is essential in various fields, including statistics, finance, and even everyday decision-making. In this comprehensive guide, we'll tackle a specific probability problem step-by-step, breaking down the concepts and formulas involved so you can confidently solve similar problems on your own. We will also explore the underlying principles of probability, such as independent and dependent events, conditional probability, and the intersection of events. By the end of this guide, you'll not only know how to find the missing probability in this particular scenario but also have a solid foundation in probability theory to tackle more complex problems. So, let's put on our thinking caps and get started! Remember, practice makes perfect, so don't hesitate to work through additional examples to solidify your understanding. Think of probability as a puzzle, and we're here to help you piece together all the elements to see the complete picture. Let’s unlock the secrets of probability together!

Let's start by stating the problem clearly. We are given the following probabilities:

• $P(A) = \frac{7}{20}$: The probability of event A occurring.
• $P(A \cap B) = \frac{49}{400}$: The probability of both events A and B occurring (the intersection of A and B).
• We need to find $P(B)$: The probability of event B occurring.

We are also provided with multiple-choice options:

• A. $\frac{3}{10}$
• B. $\frac{7}{10}$
• C. $\frac{1}{5}$
• D. $\frac{7}{20}$

This problem involves the concept of conditional probability and the relationship between the probabilities of individual events and their intersection. Before we jump into the solution, let's refresh our understanding of these concepts.

To solve this problem effectively, we need to understand the following key concepts:

1. Probability Basics

Probability is a numerical measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The probability of an event A is denoted as $P(A)$.

2. Intersection of Events

The intersection of two events, A and B, denoted as $A \cap B$, represents the event where both A and B occur simultaneously. The probability of the intersection, $P(A \cap B)$, is the likelihood of both events happening together.

3. Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$, which reads as "the probability of A given B." The formula for conditional probability is:

$P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$.

Similarly, $P(B|A) = \frac{P(A \cap B)}{P(A)}$, provided $P(A) > 0$.

This formula is crucial for solving our problem. It tells us how the probability of one event changes when we know that another event has already happened. The beauty of conditional probability lies in its ability to refine our understanding of events based on new information. In real-world scenarios, conditional probabilities help us make informed decisions, such as assessing risk or predicting outcomes based on available data. For instance, consider the probability of a patient having a certain disease given a positive test result. This involves conditional probability because the test result influences our assessment of the likelihood of the disease. The formula helps us quantify this influence and make more accurate diagnoses.

4. Rearranging the Conditional Probability Formula

We can rearrange the conditional probability formula to solve for the probability of the intersection:

$P(A \cap B) = P(B|A) \cdot P(A)$

or

$P(A \cap B) = P(A|B) \cdot P(B)$

This rearrangement is essential because it allows us to relate the probability of the intersection to the conditional probabilities and the individual probabilities of the events. In our problem, we are given $P(A)$, $P(A \cap B)$, and we need to find $P(B)$. We can use this rearranged formula to connect these probabilities and solve for the unknown.

Now that we have a solid grasp of the key concepts, let's solve the problem step by step:

1. Identify the Relevant Formula

We can use the rearranged conditional probability formula:

$P(A \cap B) = P(B|A) \cdot P(A)$

or

$P(A \cap B) = P(A|B) \cdot P(B)$

Since we want to find $P(B)$, we can rearrange the second formula to isolate $P(B)$:

$P(B) = \frac{P(A \cap B)}{P(A|B)}$

However, we don't have $P(A|B)$ directly. Instead, let's consider the other form:

$P(A \cap B) = P(A|B) \cdot P(B)$

Rearranging to solve for $P(B)$ gives us:

$P(B) = \frac{P(A \cap B)}{P(A|B)}$

But we don’t have $P(A|B)$ either. We need to find a way to relate what we have to what we need.

2. Apply the Formula

Let's use the formula:

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

We have $P(A) = \frac{7}{20}$ and $P(A \cap B) = \frac{49}{400}$. Plugging these values into the formula, we get:

$P(B|A) = \frac{\frac{49}{400}}{\frac{7}{20}}$

3. Simplify the Expression

To simplify the expression, we divide the fractions:

$P(B|A) = \frac{49}{400} \div \frac{7}{20} = \frac{49}{400} \cdot \frac{20}{7}$

Now, we can simplify by canceling out common factors:

$P(B|A) = \frac{49 \cdot 20}{400 \cdot 7} = \frac{7 \cdot 1}{20 \cdot 1} = \frac{7}{20}$

4. Identify the Correct Option

Oops! It seems we calculated $P(B|A)$ instead of $P(B)$. Let's revisit our approach. We need to use the formula:

$P(A \cap B) = P(B|A) \cdot P(A)$

We found that $P(B|A) = \frac{7}{20}$, and we know $P(A) = \frac{7}{20}$ and $P(A \cap B) = \frac{49}{400}$. We want to find $P(B)$. We can use the conditional probability formula again, but this time rearranged to solve for P(B) if we knew P(A|B).

Let’s backtrack and rethink. We have:

$P(A \cap B) = \frac{49}{400}$ $P(A) = \frac{7}{20}$

We need to find P(B). Using the formula $P(A \cap B) = P(B) \cdot P(A|B)$, we can rearrange to solve for P(B) if we knew P(A|B):

$P(B) = \frac{P(A \cap B)}{P(A|B)}$

But we don’t know $P(A|B)$. Instead, let’s look at:

$P(A \cap B) = P(A) \cdot P(B|A)$

We calculated $P(B|A) = \frac{7}{20}$. So,

$\frac{49}{400} = \frac{7}{20} \cdot P(B|A)$

This doesn't directly give us P(B). Let’s try to find a relationship that helps us.

Consider if events A and B are independent. If A and B are independent, then:

$P(A \cap B) = P(A) \cdot P(B)$

Plugging in the values:

$\frac{49}{400} = \frac{7}{20} \cdot P(B)$

Now, we can solve for $P(B)$:

$P(B) = \frac{\frac{49}{400}}{\frac{7}{20}} = \frac{49}{400} \cdot \frac{20}{7} = \frac{7}{20}$

So, $P(B) = \frac{7}{20}$.

5. Select the Correct Answer

Looking at our options, the correct answer is:

D. $\frac{7}{20}$

In this guide, we walked through a step-by-step solution to find the missing probability. We revisited the fundamental concepts of probability, including conditional probability and the intersection of events. By applying the conditional probability formula and rearranging it appropriately, we were able to determine the probability of event B, $P(B)$. Remember, the key to solving probability problems is to understand the relationships between events and apply the correct formulas. Practice with various examples to enhance your skills and build confidence in your problem-solving abilities. Keep exploring the world of probability, and you'll discover its fascinating applications in numerous real-life scenarios.

By working through this problem, we learned the importance of identifying the correct formula, rearranging it to solve for the unknown, and simplifying the expression to arrive at the final answer. We also highlighted the significance of understanding the concepts of conditional probability and the intersection of events. Probability is a powerful tool for understanding and predicting the world around us. Keep practicing, and you'll become a master of probability!

So, guys, we've successfully navigated the world of probability and found the missing piece of the puzzle! Remember, these concepts are fundamental not just in mathematics but in many real-world applications. Keep practicing, stay curious, and you'll become a probability pro in no time. Keep exploring, keep learning, and never stop questioning the world around you. Until next time, happy calculating!