Question

If two dice are thrown, then find the probability that the total score on the two dice is 5 . (1) \(2 / 9\) (2) \(3 / 8\) (3) \(1 / 6\) (4) \(1 / \overline{9}\)

Short answer

Answer: The probability is \(\frac{1}{9}\).

Step-by-step solution

Find the total number of possible outcomes

When two dice are thrown, there are 6 sides on each die, resulting in 6 x 6 = 36 different possible outcomes.

Find the number of favorable outcomes

Now, let's determine the number of ways to get a sum of 5 on both dice. There are 4 favorable outcomes: (1,4), (2,3), (3,2), and (4,1).

Calculate the probability

To find the probability, we will divide the number of favorable outcomes (4) by the total number of possible outcomes (36): \(\frac{4}{36}\) = \(\frac{1}{9}\). So, the correct answer is \(\boxed{(4) \frac{1}{9}}\).

Key concepts

Favorable Outcomes

When we talk about "favorable outcomes" in probability, we're discussing the specific situations that align with the event we're interested in. In the problem of rolling two dice and wanting a total sum of 5, the favorable outcomes are the combinations of dice rolls that add up to exactly 5. To identify these outcomes, list all possible pairs of numbers from both dice and check which ones result in the sum of 5. Here, the pairs that achieve this are:

• (1, 4): This means the first die shows a 1 and the second shows a 4.
• (2, 3): The first die shows a 2 and the second shows a 3.
• (3, 2): The first die shows a 3 and the second shows a 2.
• (4, 1): The first die shows a 4 and the second shows a 1.

These four outcomes are the only ways two six-sided dice can sum to 5. Recognizing favorable outcomes helps focus on only the scenarios that meet our specific conditions or desires.

Total Possible Outcomes

Total possible outcomes help us understand the complete sample space of an event. For two six-sided dice, each die has 6 faces. When rolled together, the first die can land on any of its 6 faces, while the second die, simultaneously, also has 6 options. These two independent events multiply to give the total number of possible outcomes.
By multiplying the number of outcomes of one die by the number of outcomes of the other die, we find:\[6 \times 6 = 36\]This means there are 36 different possible outcomes or combinations when rolling two dice. Understanding the total possible outcomes gives us a denominator to calculate probability, as it encompasses all possible results, whether they're favorable or not.

Sum of Dice Roll

The sum of a dice roll is simply the total of the numbers showing on the top faces after the dice have been thrown. When rolling two dice, this sum can range from a minimum of 2 (if both dice land on 1) to a maximum of 12 (if both show a 6). In our specific problem, we're interested in the sum being exactly 5. To achieve a sum of 5, we check different combinations of numbers on the dice. Each die contributes to the total, and only specific pairs (like 1+4 or 2+3) will give the desired sum. It's important to understand each combination:

• If the first die shows a 1, the second must show a 4.
• If the first die shows a 2, the second must show a 3.
• If the first die shows a 3, the second must show a 2.
• If the first die shows a 4, the second must show a 1.

Understanding the sum of dice roll helps in calculating specific probabilities in dice games, forming a crucial part of assessing potential outcomes.