Problem 43 - Algebra and Trigonometry

Question

Two fair dice are rolled. Determine the probability that the sum of the faces is 3 .

Step-by-Step Solution

Answer

The probability is \( \frac{1}{18} \).

Step 1 - Identify Total Possible Outcomes

When rolling two six-sided dice, each die has 6 faces. Therefore, the total number of possible outcomes is calculated by multiplying the number of faces on the first die by the number of faces on the second die. Thus, the total number of possible outcomes is: \(6 \times 6 = 36\)

Step 2 - List Favorable Outcomes

To find the probability of rolling a sum of 3, identify the pairs of dice that add up to 3. These pairs are: (1, 2) and (2, 1). These are the favorable outcomes.

Step 3 - Count the Favorable Outcomes

Count the number of favorable outcomes. From the previous step, we see there are 2 such outcomes: (1, 2) and (2, 1).

Step 4 - Calculate the Probability

The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. Thus, the probability \(P\) that the sum of the faces is 3 is:\[ P = \frac{ \text{Number of Favorable Outcomes} }{ \text{Total Number of Possible Outcomes} } = \frac{2}{36} = \frac{1}{18} \]

Key Concepts

probability calculation

When we talk about probability, we are referring to the chance of a specific event occurring out of all possible outcomes. It's like asking how likely something is to happen.
To calculate this, we use a simple formula: \[ Probability = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \] In the case of rolling two dice, our total number of possible outcomes is easy to determine. Each die has 6 faces, so when we roll two dice, we have a total of \(6 \times 6 = 36\) possible outcomes.
This total is essential for calculating probabilities because we compare the number of ways the desired event can happen (favorable outcomes) to this total number of possible outcomes. The fraction we get from this comparison gives us the probability.

dice outcomes

Rolling two six-sided dice is a classic example used to understand probability. Each die has faces numbered from 1 to 6.
When we roll two dice, each face of the first die can land with any face of the second die. This means we have 36 unique outcomes.
Some important facts about dice outcomes:

• Each specific outcome, like rolling a 1 on the first die and a 2 on the second die, is unique.
• These outcomes are equally likely, meaning each has the same chance of occurring.

For example, if we want the two dice to sum up to 3, we need to look for all pairs that result in this sum. Possible pairs are:

• (1,2)
• (2,1)

These are our 'favorable outcomes' since they meet the condition we are interested in.

favorable outcomes

Favorable outcomes are the specific outcomes we are interested in for a given probability problem.
Imagine we want the sum of two dice to be 3. The favorable outcomes in this case are the pairs (1, 2) and (2, 1). These pairs add up to 3, which makes them favorable.
It's important to identify all such outcomes to correctly calculate probability. Here’s how:

• First, define the condition of interest (e.g., summing to 3).
• Next, list all outcomes that meet this condition.

Once we have this list, we count the number of favorable outcomes and use it in our probability formula.
In our example, there are 2 favorable outcomes out of 36 possible outcomes. Using our formula, the probability is given by \[ \frac{2}{36} = \frac{1}{18} \] Understanding favorable outcomes is crucial since it directly impacts the result of our probability calculation.