Problem 41 - Algebra and Trigonometry

Question

Two fair dice are rolled. Determine the probability that the sum of the faces is \(7 .\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{6} \).

Step 1: Identify Total Possible Outcomes

When two fair dice are rolled, each die has 6 faces. Thus, the total number of possible outcomes is calculated by multiplying the number of faces on each die: \[ 6 \times 6 = 36 \].

Step 2: List the Successful Outcomes

Next, identify the outcomes where the sum of the faces equals 7. The possible pairs that sum to 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Therefore, there are 6 successful outcomes.

Step 3: Calculate the Probability

To find the probability, divide the number of successful outcomes by the total number of possible outcomes: \[ P(\text{Sum of 7}) = \frac{6}{36} = \frac{1}{6} \].

Key Concepts

probability

Probability helps us understand how likely an event is to occur. It is a measure between 0 and 1.
If the probability is 0, the event will never happen. If it is 1, the event will always happen.
When calculating probability, we often use the formula: \( P(\text{Event}) = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}} \).
This formula works for any event, like rolling dice in our example.

dice outcomes

When rolling two fair dice, each with 6 faces, the possible outcomes include all the combinations of the faces.
There are 36 outcomes in total, calculated by multiplying the number of faces on each die, \(6 \times 6 = 36\).
Some examples of possible outcomes are (1,1), (2,3), and (6,6). Each combination is equally likely to occur in fair dice rolls.

sum of numbers

Now, let’s identify all outcomes that result in a sum of 7:
The pairs that give a sum of 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
These represent the successful outcomes for our event, leading to 6 successful outcomes in total.
The formula we learned earlier helps us find the probability of any sum.

fair dice

Fair dice are unbiased and have an equal probability of landing on any face.
Each face of the die shows up about 1/6 of the time in a large number of rolls.
This fairness is crucial for calculating probabilities, as each outcome is equally likely.
When using fair dice, probabilities reflect true randomness of the roll, making calculations like the sum of 7 accurate and reliable.