Probability Of B: A Step-by-Step Solution

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Probability of B: A Step-by-Step Solution

Let's dive into this probability problem together, guys! We're given some probabilities related to events A and B, and our mission is to find the probability of event B occurring. It sounds like a fun challenge, right? To solve this, we'll need to use some fundamental concepts of probability theory, including the inclusion-exclusion principle and the complement rule. Don't worry if these terms sound intimidating; we'll break everything down step by step. So, let's put on our thinking caps and get started!

Understanding the Problem

First, let's clearly define what we know. We're given the following probabilities:

  • P(A) = 0.5: This means the probability of event A occurring is 0.5, or 50%.
  • P(A ∩ B) = 0.35: This is the probability of both events A and B occurring simultaneously. The symbol "∩" represents the intersection of events, meaning the outcomes that belong to both A and B.
  • P(neither A nor B occurs) = 0.4: This tells us the probability that neither event A nor event B happens. In probability terms, this is the probability of the complement of the union of A and B. We'll represent this as P((A∪B)c)=0.4P((A ∪ B)^c) = 0.4, where "∪" means the union of events (A or B or both), and "c" denotes the complement.

Our ultimate goal is to find P(B), the probability of event B occurring.

Key Takeaway: Make sure you understand what each probability represents. Visualizing this with a Venn diagram can be super helpful! Event A is 50% likely. Both events, A and B, happen 35% of the time. The chance of neither A nor B happening is 40%. Now, let’s use this information to find out how likely just B is.

Applying Probability Principles

To find P(B), we'll need to use a couple of key probability principles. The first one is the inclusion-exclusion principle, which is a fancy name for a pretty intuitive idea. It helps us calculate the probability of the union of two events (A or B). The principle states:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

This formula makes sense if you think about it. If we simply add P(A) and P(B), we're double-counting the outcomes that belong to both A and B (the intersection). So, we need to subtract P(A ∩ B) to correct for this overcounting.

The second principle we'll use is the complement rule. This rule states that the probability of an event not occurring (its complement) is equal to 1 minus the probability of the event occurring:

P(E^c) = 1 - P(E)

where E is any event, and EcE^c is its complement.

In our case, we know P((A∪B)c)=0.4P((A ∪ B)^c) = 0.4. So, we can use the complement rule to find P(A∪B)P(A ∪ B): P(A∪B)=1−P((A∪B)c)=1−0.4=0.6P(A ∪ B) = 1 - P((A ∪ B)^c) = 1 - 0.4 = 0.6

Key Takeaway: The inclusion-exclusion principle helps us avoid double-counting when finding the probability of A or B, and the complement rule links the probability of an event happening to the probability of it not happening. So, the chance of either A or B happening is 60% because the chance of neither happening is 40%. This is a critical piece of the puzzle! Now we're one step closer to figuring out P(B).

Solving for P(B)

Now we have all the pieces we need to solve for P(B)! We know:

  • P(A ∪ B) = 0.6
  • P(A) = 0.5
  • P(A ∩ B) = 0.35

We can plug these values into the inclusion-exclusion principle:

  1. 6 = 0.5 + P(B) - 0.35

Now, it's just a matter of solving for P(B). Let's simplify the equation:

  1. 6 = 0.15 + P(B)

Subtract 0.15 from both sides:

P(B) = 0.6 - 0.15

P(B) = 0.45

Therefore, the probability of event B occurring is 0.45, or 45%.

Key Takeaway: By carefully applying the inclusion-exclusion principle and the complement rule, we were able to isolate P(B) and calculate its value. This demonstrates the power of these fundamental probability concepts. Guys, we nailed it! The probability of event B happening is 45%. This is our final answer, and we got there by using the principles of probability we discussed.

Visualizing with a Venn Diagram (Optional)

If you're a visual learner, a Venn diagram can be a super helpful tool for understanding this problem. Imagine two overlapping circles, one representing event A and the other representing event B. The overlapping region represents the intersection (A ∩ B).

  • The area of the circle representing A is 0.5.
  • The area of the overlapping region (A ∩ B) is 0.35.
  • The area outside both circles (neither A nor B) is 0.4.

By filling in the areas based on the given probabilities, you can visually see how the different probabilities relate to each other and how P(B) can be calculated.

Key Takeaway: Venn diagrams can provide a visual representation of probability problems, making them easier to understand and solve. Seeing the overlapping areas helps to avoid confusion with double-counting and makes the problem more intuitive. If you are more of a visual person, try drawing this out—it really helps!.

Conclusion

So, guys, we've successfully solved this probability problem! We were able to find P(B) by applying the inclusion-exclusion principle and the complement rule. Remember, probability problems often require breaking down the information into smaller parts and using the appropriate formulas and principles.

The key steps we followed were:

  1. Understanding the given information: Clearly identify what probabilities are given and what we need to find.
  2. Applying the complement rule: Use the given probability of