Problem 41 - College Algebra

Question

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum of 5 or 6 .

Step-by-Step Solution

Answer

The probability of rolling a sum of 5 or 6 is 1/4.

Step 1: Understanding Dice Roll Outcomes

When two dice are rolled, each die has 6 possible outcomes (1 through 6). Therefore, there are a total of 6 x 6 = 36 different possible outcomes for the roll of two dice. Each outcome is equally likely.

Step 2: Identifying Favorable Outcomes for a Sum of 5

To have a sum of 5, the following pairs of dice must be rolled: (1,4), (2,3), (3,2), or (4,1). There are 4 combinations that result in a sum of 5.

Step 3: Identifying Favorable Outcomes for a Sum of 6

For a sum of 6, the pairs of dice needed are: (1,5), (2,4), (3,3), (4,2), or (5,1). This gives us 5 different combinations that sum to 6.

Step 4: Calculating Total Favorable Outcomes

Add the number of favorable outcomes for both sums. There are 4 outcomes for a sum of 5 and 5 outcomes for a sum of 6, totalling to 4 + 5 = 9 favorable outcomes.

Step 5: Calculating Probability of Rolling a 5 or 6

The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Therefore, the probability of rolling a sum of 5 or 6 is 9/36, which simplifies to 1/4.

Key Concepts

Dice Probabilities

When it comes to understanding the probabilities associated with dice, it's essential to start with the basics. Each die in a pair has six faces, numbered 1 through 6. When rolling two dice together, you increase the range of possible outcomes. Every roll of two dice results in a combination of the two numbers, leading to a total of 36 possible outcomes. Imagine each die independently, and you'll realize that each outcome is equally likely. This is because the result of one die does not influence the other die. Given that each combination is equally possible, we say that each outcome has a probability of 1/36.

Sum of Two Dice

One intriguing aspect of rolling two dice is calculating the likelihood of various sums. The smallest possible sum is 2 (1 + 1), and the largest is 12 (6 + 6). Between these extremes, you can achieve sums like 5 or 6, which are the sums we are interested in for this problem. Each sum is the result of different combinations of the values on both dice. For instance:

• To get a sum of 5, you might roll (1,4), (2,3), (3,2), or (4,1).
• For a sum of 6, you could toss a (1,5), (2,4), (3,3), (4,2), or (5,1).

Each possible sum corresponds to a unique set of combinations.

Favorable Outcomes

In probability, 'favorable outcomes' are those outcomes which satisfy the conditions of the event we're considering. For the problem we have, a 'favorable outcome' is any roll of two dice that results in a sum of either 5 or 6.
We have already identified specific combinations that produce these sums.

• There are 4 favorable outcomes resulting in a 5.
• 5 different combinations yield a sum of 6.

Together, these give us 9 favorable outcomes that satisfy our condition. Recognizing these combinations is crucial for calculating the probability of events in dice games.

Calculating Probability

Probability in mathematics measures how likely an event is to occur.
The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In the dice example, once we've determined there are 9 favorable outcomes for rolling a sum of 5 or 6, we see:

• Total possible outcomes = 36 (from rolling two dice)
• Favorable outcomes = 9 (combinations yielding 5 or 6)

The probability formula thus becomes: \[\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{9}{36} = \frac{1}{4}\]This means the probability of getting a sum of 5 or 6 when rolling two dice is 1 out of 4, or 25%. Understanding how to calculate this probability provides a firm foundation for more complex statistical concepts.