Understanding Probability with Two Dice

When we throw two dice simultaneously, each die has 6 possible outcomes (1, 2, 3, 4, 5, 6). Since the outcome of one die does not affect the outcome of the other, the total number of possible outcomes when throwing two dice is the product of the number of outcomes for each die.

Sample Space for Two Dice

The total number of possible outcomes, also known as the size of the sample space, is calculated as:

\( \text{Total Outcomes} = \text{Outcomes on Die 1} \times \text{Outcomes on Die 2} \)

\( \text{Total Outcomes} = 6 \times 6 = 36 \)

These 36 outcomes are pairs of numbers representing the result on the first die and the second die (e.g., (1,1), (1,2), ..., (6,6)).

Identifying Favorable Outcomes

We are interested in the probability of getting a sum of 2, 8, or 12. We need to identify the specific outcomes from the sample space that result in these sums.

Outcomes with a Sum of 2:

The only pair of numbers that adds up to 2 is (1, 1).

• (1, 1)

Number of outcomes with a sum of 2: 1

Outcomes with a Sum of 8:

The pairs of numbers that add up to 8 are:

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

Number of outcomes with a sum of 8: 5

Outcomes with a Sum of 12:

The only pair of numbers that adds up to 12 is (6, 6).

• (6, 6)

Number of outcomes with a sum of 12: 1

The events of getting a sum of 2, a sum of 8, and a sum of 12 are mutually exclusive because they cannot happen at the same time on a single throw of two dice.

Calculating Total Favorable Outcomes

Since the events (sum 2, sum 8, sum 12) are mutually exclusive, the total number of favorable outcomes for the event "sum 2 or 8 or 12" is the sum of the number of outcomes for each individual event.

\( \text{Total Favorable Outcomes} = (\text{Outcomes for Sum 2}) + (\text{Outcomes for Sum 8}) + (\text{Outcomes for Sum 12}) \)

\( \text{Total Favorable Outcomes} = 1 + 5 + 1 = 7 \)

Calculating the Probability

The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

\( \text{Probability} = \frac{\text{Total Favorable Outcomes}}{\text{Total Possible Outcomes}} \)

\( \text{Probability (Sum 2 or 8 or 12)} = \frac{7}{36} \)

Therefore, the probability of getting a sum of 2 or 8 or 12 when throwing two dice simultaneously is \( \frac{7}{36} \).

Revision Table: Two Dice Probability Concepts

Additional Information: Probability Basics

Probability is a measure of how likely an event is to occur. It is a value between 0 and 1, inclusive. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain to happen.

When dealing with multiple events, understanding whether they are mutually exclusive or independent is crucial for calculating combined probabilities.

• Mutually Exclusive Events: If A and B are mutually exclusive, the probability of A or B occurring is \(P(A \cup B) = P(A) + P(B)\). This is what we used in this problem.
• Independent Events: If A and B are independent, the probability of A and B occurring is \(P(A \cap B) = P(A) \times P(B)\). Rolling two dice are examples of independent events (the outcome of one die does not influence the other), but the question asks about sums which are dependent on both outcomes. The events of getting sum 2, sum 8, and sum 12 on a single throw are mutually exclusive.

The problem requires identifying all outcomes that meet the conditions (sum 2, sum 8, or sum 12) and dividing by the total number of possible outcomes (the size of the sample space).