Probability Of Sum ≤ 9 When Rolling 2 Dice Given 1st Dice Is 5

Hey guys! Let's dive into a super interesting probability problem. We're going to figure out the probability of getting a sum of 9 or less when rolling two dice, but with a twist – we already know the first die landed on a 5. This is a classic example of conditional probability, and it’s way easier to tackle than it might sound at first. So, grab your thinking caps, and let’s roll!

Understanding Conditional Probability

Before we jump into the dice, let’s quickly chat about conditional probability. In simple terms, conditional probability is the chance of something happening given that something else has already happened. It’s like saying, "What's the probability of it raining tomorrow, given that it’s cloudy today?" The "given" part is super important because it changes the whole game. We are not looking at all possible outcomes anymore, but only at the outcomes that satisfy the condition.

In math terms, we write conditional probability as P(A|B), which we read as "the probability of A given B." Here, A is the event we’re interested in (like the sum of the dice being ≤ 9), and B is the event we know has already happened (the first dice is a 5). The formula for conditional probability is:

P(A|B) = P(A and B) / P(B)

Where:

• P(A|B) is the probability of event A happening given that event B has happened.
• P(A and B) is the probability of both events A and B happening together.
• P(B) is the probability of event B happening.

For our dice problem, this means we need to find the probability of the sum being ≤ 9 and the first die being a 5, and then divide that by the probability of the first die being a 5. Sounds like a plan, right? Let’s break it down step by step.

Breaking Down the Dice Problem

Okay, let's get specific about our dice. We're rolling two standard six-sided dice. The first crucial piece of information is that the first die is a 5. This is our given condition. So, we don’t need to think about all 36 possible outcomes (6 sides on the first die times 6 sides on the second die). Instead, we only need to consider the outcomes where the first die shows a 5. This significantly narrows down our possibilities and makes the problem much more manageable. Remember, the probability we're after is the probability that the sum of both dice is less than or equal to 9. This is our target event. To solve this, we'll list out all the outcomes where the first die is a 5, and then identify which of these outcomes also have a sum less than or equal to 9. This will give us the number of favorable outcomes for our conditional probability calculation.

When the first die is a 5, the possible outcomes are:

• (5, 1)
• (5, 2)
• (5, 3)
• (5, 4)
• (5, 5)
• (5, 6)

These are all the possibilities when the first die shows a 5. There are six outcomes in total. Now, let's figure out which of these outcomes result in a sum that's less than or equal to 9. This is the next key step in calculating our conditional probability.

Identifying Favorable Outcomes

Alright, let's roll up our sleeves and figure out which of those outcomes give us a sum of 9 or less. Remember, we already know the first die is a 5, so we just need to look at the second die and see how it affects the total. This is where the core of our problem-solving lies. By identifying these favorable outcomes, we're essentially figuring out the numerator of our conditional probability fraction – the probability of both events happening together.

Let's go through our list:

• (5, 1): 5 + 1 = 6 (≤ 9) – This one's a winner!
• (5, 2): 5 + 2 = 7 (≤ 9) – Another good one!
• (5, 3): 5 + 3 = 8 (≤ 9) – We're on a roll!
• (5, 4): 5 + 4 = 9 (≤ 9) – Just made the cut!
• (5, 5): 5 + 5 = 10 ( > 9) – Nope, too high.
• (5, 6): 5 + 6 = 11 ( > 9) – Definitely too high.

So, we have four favorable outcomes: (5, 1), (5, 2), (5, 3), and (5, 4). These are the only combinations where the first die is a 5 and the sum of the two dice is 9 or less. Now we have a clear picture of the outcomes that satisfy both conditions, which is crucial for calculating our final probability.

Calculating the Probability

Okay, we've done the hard work of identifying our favorable outcomes. Now comes the fun part – putting it all together to calculate the probability. We're going to use our understanding of conditional probability to get to the final answer. Remember the formula we talked about earlier?

P(A|B) = P(A and B) / P(B)

In our case:

• A is the event that the sum of the two dice is ≤ 9.
• B is the event that the first die is a 5.

So, P(A|B) is the probability that the sum is ≤ 9 given that the first die is a 5.

Let’s break down the parts we need:

• P(A and B): This is the probability that the sum is ≤ 9 and the first die is a 5. We found 4 favorable outcomes out of a total of 36 possible outcomes when rolling two dice. However, since we already know the first die is a 5, we are only considering the 6 outcomes where the first die is a 5. Out of these 6 outcomes, 4 have a sum ≤ 9. So, the probability is 4/6.
• P(B): This is the probability that the first die is a 5. Since there are 6 sides on a die, the probability of rolling a 5 on the first die is 1/6. However, in this conditional probability problem, we are given that the first die is a 5, so we only consider outcomes where the first die is a 5. This means we are working within a reduced sample space.

Now, let's plug these values into our formula:

To find P(A|B), we need to consider the reduced sample space where the first die is a 5. There are 6 possible outcomes in this space: (5,1), (5,2), (5,3), (5,4), (5,5), and (5,6). We found that 4 of these outcomes have a sum ≤ 9. Therefore, the conditional probability is:

P(A|B) = (Number of favorable outcomes) / (Total possible outcomes given the condition)

P(A|B) = 4 / 6

Simplifying the Fraction

We're almost there! We've got the probability as 4/6, but remember, we need to express our answer as a fraction in its simplest form. This is the final touch that makes our answer perfect. Simplifying fractions is like giving our answer a little polish to make it shine.

To simplify 4/6, we need to find the greatest common divisor (GCD) of 4 and 6. The GCD is the largest number that divides both numbers evenly. In this case, the GCD of 4 and 6 is 2.

Now, we divide both the numerator (4) and the denominator (6) by the GCD (2):

4 ÷ 2 = 2

6 ÷ 2 = 3

So, the simplified fraction is 2/3. This is our final answer! We've taken the initial probability of 4/6 and made it as clear and concise as possible.

The Final Answer

Drumroll, please! The probability that the sum of the 2 dice is ≤ 9, given that the 1st dice is a 5, is 2/3. Ta-da! We did it!

Isn't it awesome how we can break down a seemingly complex problem into smaller, manageable steps? We started by understanding conditional probability, then we carefully identified our favorable outcomes, calculated the probability, and finally, simplified our fraction. Each step was crucial in getting us to the right answer. This problem not only tests your understanding of probability but also highlights the importance of clear, logical thinking. Remember, probability problems often seem daunting, but with a systematic approach, you can conquer them all. Keep practicing, and you’ll become a probability pro in no time!

I hope you guys enjoyed this breakdown. Keep those dice rolling and those brains churning!