Problem 38 - College Algebra

Question

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum less than 15 .

Step-by-Step Solution

Answer

The probability is 1.

Step 1: Determine Total Possible Outcomes

When we roll two dice, each die has 6 faces, numbered from 1 to 6. Therefore, the total number of possible outcomes when rolling two dice is calculated as follows:\[ 6 \times 6 = 36 \] This means there are 36 possible outcomes when rolling two dice.

Step 2: Identify Favorable Outcomes for Sums Less Than 15

We need to find all possible sums of the two dice that are less than 15. The possible sums when two dice are rolled range from 2 (1+1) to 12 (6+6). Since all of these sums are less than 15, every sum is favorable. There are no sums of 15 or more when two standard 6-sided dice are rolled.

Step 3: Calculate Number of Favorable Outcomes

Since every outcome is favorable (as calculated in Step 2), the number of favorable outcomes is equal to the total number of outcomes, which is 36.

Step 4: Calculate the Probability

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. Here, both are 36:\[ \frac{36}{36} = 1 \]This means the probability of rolling a sum less than 15 on two dice is 1.

Key Concepts

Two Dice Roll

Rolling two dice simultaneously involves a fundamental aspect of probability theory. When you roll a single die, it has 6 faces, each numbered from 1 to 6. Thus, when two dice are rolled, you need to consider all combinations of the numbers on each die. These combinations are referred to as outcomes.

• Each die is independent of the other. This means the result on one die does not affect the result on the other die.
• The total number of outcomes is found by multiplying the number of outcomes for each die together: \(6 \times 6 = 36\).

Hence, there are 36 possible outcomes when two dice are rolled together.

Sum of Dice

The sum of the two dice refers to adding the numbers that appear on the top face of each die after a roll. This sum is a key factor in many dice games and probability problems.

• The smallest sum occurs when both dice show 1, totaling 2 (i.e., \(1 + 1 = 2\)).
• The largest sum occurs when both dice show 6, totaling 12 (i.e., \(6 + 6 = 12\)).

Therefore, the possible sums range from 2 to 12. In this specific exercise, we are interested in sums less than 15, which naturally includes all these possible sums.

Probability Calculation

Probability is a measure of how likely an event is to occur. It's calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
For instance, if we want to know the probability of a certain sum appearing when rolling two dice, we need:

• Favorable outcomes: the sums that meet the criteria (e.g., less than 15).
• Total possible outcomes: always 36 when rolling two dice.

The formula for probability is given by \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} \). In this case, it's \( \frac{36}{36} = 1 \), since all possible outcomes (sums) are favorable.

Favorable Outcomes

In probability, favorable outcomes are the specific results that satisfy the condition we are interested in. For this exercise, we want sums less than 15. These sums include all possible totals from rolling two dice, ranging from 2 to 12.

• Every possible combination resulting from a roll of two 6-sided dice gives a sum within this range.
• Since all possible sums of two dice are less than 15, each outcome is favorable.

Thus, the number of favorable outcomes equals the total outcomes, 36, leading to a probability of 1 for rolling a sum less than 15.