Problem 35 - Algebra 2

Question

Determine whether each probability is theoretical or experimental. Then find the probability. Two dice are rolled. What is the probability that the sum will be 12?

Step-by-Step Solution

Answer

Theoretical probability is \( \frac{1}{36} \).

Step 1: Understanding the Scenario

When we roll two dice, each die has 6 faces with numbers from 1 to 6. We want to determine the probability that the sum of the numbers shown on the two dice is 12. This is a theoretical probability calculation because we are considering all possible outcomes.

Step 2: Calculating Total Possible Outcomes

To find the total number of possible outcomes when rolling two dice, consider each die individually. For each of the 6 outcomes of the first die, there are 6 outcomes for the second die. Thus, the total number of possible outcomes is: \( 6 \times 6 = 36 \).

Step 3: Identifying Favorable Outcomes

Next, we find the outcomes where the sum is 12. This occurs only when both dice show a 6. So, there is exactly 1 favorable outcome: (6,6).

Step 4: Calculating Theoretical Probability

The theoretical probability of the event (sum of 12) is the ratio of the number of favorable outcomes to the total number of possible outcomes. Thus, the probability is: \( \frac{1}{36} \).

Key Concepts

Experimental Probability

Experimental probability is about observing the likelihood of an event occurring through trials or experiments. Unlike theoretical probability which uses a well-defined formula, experimental probability relies on actual results. Suppose you roll two dice several times, and each time you record the outcomes. The experimental probability would be calculated by dividing the number of times you observe the sum as 12 by the total number of trials. For example:

• If you roll the dice 100 times, and the pair (6,6) appears 2 times, the experimental probability of getting a sum of 12 is \ \( \frac{2}{100} = 0.02 \) or 2\%.

Unlike theoretical probability, experimental results may vary each time due to different numbers of trials and random chance.

Favorable Outcomes

Favorable outcomes are those specific results in an experiment or calculation that align with the desired event. In our scenario, we're eager to know when the sum of numbers on two dice equals 12.
tIn the world of dice, each roll represents a separate outcome. For our specific case:

• When rolling two six-sided dice, the only favorable event for obtaining a sum of 12 is when both dice show a 6, making it a single favorable outcome.

Understanding favorable outcomes helps us to identify the unique conditions required to achieve the event we are analyzing.

Total Possible Outcomes

Total possible outcomes represent all the different results that can occur in a given scenario. To determine this for rolling two dice, remember that each die has 6 faces. Each face represents a possible outcome. When calculating the number of possible outcomes, consider each die separately:

• The first die can exhibit any one of 6 numbers.
• The second die, also exhibiting any one of 6 numbers, combines with the first die's numbers.

Mathematically, this results in a total of \ \( 6 \times 6 = 36 \) different combinations. Knowing the total number of possible outcomes provides the basis for determining probability.

Dice Probability

Dice probability is a form of theoretical probability that calculates the likelihood of specific outcomes when dice are rolled. Dice are particularly interesting in probability because they have a fixed number of sides, usually six. For our problem, we calculated the probability of achieving a sum of 12 after rolling two dice. Theoretical probability can always be calculated with the formula:\[\text{Probability (P)} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}}\]Let's apply this formula:

• Favorable outcomes for a sum of 12: 1 (one pair of dice: 6,6).
• Total possible outcomes when rolling two dice: 36.
• Thus, probability (sum of 12) = \ \( \frac{1}{36} \) or about 2.78%.

This fundamental concept helps build a strong foundation in understanding probabilities across all kinds of dice scenarios.