(II) Calculate the probabilities, when you throw two dice, of obtaining a 7.

Verified step by step guidance

1. Understand the problem: When throwing two dice, each die has 6 faces numbered from 1 to 6. The total number of possible outcomes is 6 × 6 = 36, as each die is independent of the other.
2. Identify the favorable outcomes: To obtain a sum of 7, the possible pairs of numbers on the two dice are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). This gives a total of 6 favorable outcomes.
3. Calculate the probability: The probability of an event is given by the formula: $\frac{\mathrm{favorable} \mathrm{outcomes}}{\mathrm{total} \mathrm{outcomes}}$. Substituting the values, the probability is $\frac{6}{36}$.
4. Simplify the fraction: Simplify $\frac{6}{36}$ to its lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 6.
5. Interpret the result: The simplified probability represents the likelihood of obtaining a sum of 7 when throwing two dice. This is the final step in solving the problem.

Key Concepts

Probability

Probability is a measure of the likelihood of an event occurring, expressed as a ratio of favorable outcomes to the total number of possible outcomes. In the context of rolling dice, it quantifies how likely it is to achieve a specific sum, such as 7, when two dice are thrown.

Sample Space

The sample space is the set of all possible outcomes of a random experiment. For two six-sided dice, the sample space consists of 36 outcomes, represented by the ordered pairs (1,1) through (6,6). Understanding the sample space is crucial for calculating probabilities accurately.

Favorable Outcomes

Favorable outcomes refer to the specific results that satisfy the condition of interest—in this case, obtaining a sum of 7 when rolling two dice. There are six combinations that yield a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Identifying these outcomes is essential for calculating the probability.