Probability: Yellow Or Even Ticket?

Nov 18, 2025 by Admin 36 views

Let's break down this probability problem step by step, guys! We've got a bag filled with tickets, some yellow and some green, each with a number on it. The question is: what's the chance of grabbing a ticket that's either yellow or has an even number on it? It sounds like a fun little puzzle, so let's get started!

Understanding the Problem

First, let's lay out all the information we have:

Yellow Tickets: There are three yellow tickets, numbered 1, 2, and 3.
Green Tickets: There are four green tickets, numbered 1, 2, 3, and 4.

Our goal is to find the probability of picking a ticket that satisfies either of these conditions:

The ticket is yellow.
The ticket has an even number.

Remember, probability is calculated as:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

So, we need to figure out the total number of possible outcomes (how many tickets are there in total?) and the number of favorable outcomes (how many tickets are yellow or have an even number?).

Calculating Total Possible Outcomes

This part is pretty straightforward. We simply add up the number of yellow and green tickets.

Total tickets = 3 yellow tickets + 4 green tickets = 7 tickets

So, there are 7 possible outcomes when we pick a ticket randomly. This will be the denominator in our probability fraction.

Calculating Favorable Outcomes

This is where things get a little more interesting. We need to count the tickets that are either yellow or have an even number. Let's list them out:

Yellow Tickets: 1, 2, 3 (all three are favorable because they are yellow)
Even-Numbered Tickets: 2, 4 (2 is already counted in the yellow tickets, but 4 is a new favorable outcome from the green tickets)

So, our favorable outcomes are: 1, 2, 3 (yellow) and 4 (green, even). That gives us a total of 4 favorable outcomes.

It's super important to avoid double-counting! Notice that the ticket numbered '2' is both yellow and even. We've already counted it once as a yellow ticket, so we don't count it again when considering even-numbered tickets. This is a crucial point in probability problems involving "or" conditions.

To make it even clearer, we can use the principle of inclusion-exclusion:

Number of (Yellow or Even) = Number of Yellow + Number of Even - Number of (Yellow and Even)

Number of Yellow = 3
Number of Even = 2 (tickets 2 and 4)
Number of (Yellow and Even) = 1 (ticket 2)

So, Number of (Yellow or Even) = 3 + 2 - 1 = 4

This confirms that we have 4 favorable outcomes.

Calculating the Probability

Now we have all the pieces we need! We know:

Number of favorable outcomes = 4
Total number of possible outcomes = 7

Therefore, the probability of picking a yellow ticket or an even-numbered ticket is:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 4/7

Wait a minute! Looking back at the options, 4/7 isn't one of them. Let's re-examine everything to make sure we didn't make a mistake. The yellow tickets are 1, 2, and 3. The green tickets are 1, 2, 3, and 4. The tickets that are yellow or even are 1, 2, 3, and 4. Thus, there are 5 favorable outcomes, not 4. So the probability should be 5/7.

Let's Go Through Another Example

Okay, let's solidify our understanding with a slightly different example. Suppose we have a deck of cards with only the numbers 1 through 5. What's the probability of drawing a card that's either odd or greater than 3?

Odd Cards: 1, 3, 5
Cards Greater Than 3: 4, 5

Favorable Outcomes: 1, 3, 4, 5 (Notice that 5 is both odd and greater than 3, but we only count it once). Total Possible Outcomes: 5 (since there are 5 cards) Probability = 4/5

See how we carefully considered the overlap to avoid double-counting? This is the key to mastering these types of probability problems. It requires attention to detail and a systematic approach.

Common Mistakes to Avoid

Here are some common pitfalls people encounter when tackling probability problems involving "or":

Double-Counting: This is the biggest one! Always, always check for overlap between the categories you're considering and make sure you're not counting the same outcome multiple times.
Misunderstanding "Or": Remember that "or" in probability means either one condition is true, or the other condition is true, or both conditions are true. It's inclusive.
Incorrectly Identifying Outcomes: Make sure you correctly list all the possible outcomes and all the favorable outcomes. A simple mistake in listing can throw off your entire calculation.
Forgetting the Basics: Don't forget the fundamental definition of probability: (Favorable Outcomes) / (Total Possible Outcomes).

Tips for Solving Probability Problems

Here's a summary of helpful strategies for solving these types of problems:

Read Carefully: Understand the problem statement completely before you start calculating anything.
List Outcomes: Write out all possible outcomes and all favorable outcomes. This helps prevent errors.
Check for Overlap: Be vigilant about identifying and avoiding double-counting.
Use Inclusion-Exclusion (if needed): If you're struggling to account for overlap, the principle of inclusion-exclusion can be a helpful tool.
Simplify: If possible, simplify the problem by breaking it down into smaller, more manageable steps.
Practice: The more probability problems you solve, the better you'll become at recognizing patterns and avoiding common mistakes.

Conclusion

Probability problems can seem tricky at first, but with a clear understanding of the basic principles and a systematic approach, you can conquer them! Remember to carefully define the problem, identify all possible and favorable outcomes, avoid double-counting, and practice, practice, practice. With these tools in your arsenal, you'll be well on your way to mastering probability!

In our original problem, we determined that the probability of picking a yellow ticket or an even-numbered ticket is 5/7. So, the correct answer is C. 5/7