Problem 43 - Algebra for College Students

Question

Two dice are tossed 720 times. How many times would you expect to get a sum greater than 9 ? 120

Step-by-Step Solution

Answer

Expect a sum greater than 9 about 120 times in 720 tosses.

Step 1: Determine Possible Outcomes

When two dice are tossed, each die has 6 faces, resulting in a total of 6 x 6 = 36 possible outcomes. Each pair (such as (1,1), (1,2), etc.) represents an outcome of the two-dice toss.

Step 2: Identify Outcomes with Sum Greater than 9

We need to count the pairs of dice rolls that sum to a number greater than 9. These sums are 10, 11, and 12. Possible pairs are: - Sum 10: (4,6), (5,5), (6,4) - Sum 11: (5,6), (6,5) - Sum 12: (6,6) This totals to 3 + 2 + 1 = 6 outcomes.

Step 3: Calculate Probability of Desired Outcomes

The probability of getting a sum greater than 9 in a single roll is the number of favorable outcomes divided by the total outcomes. Thus, the probability is \( \frac{6}{36} = \frac{1}{6} \).

Step 4: Compute Expected Frequency

To find the expected frequency of getting a sum greater than 9 when tossing the dice 720 times, multiply the probability by the number of trials (720). Expected frequency = \( \frac{1}{6} \times 720 = 120 \).

Key Concepts

Expected Frequency

Expected Frequency is a concept used in probability to predict how often a particular event is likely to occur within a set number of trials. When we talk about expected frequency, we're essentially looking for an average or mean count of successful outcomes over multiple opportunities. Here’s how it works:

• First, calculate the probability of the event happening in one trial.
• Multiply that probability by the total number of trials being conducted.

In our dice game, tossing two dice 720 times, we're interested in how frequently we expect to achieve a sum greater than 9. Using the probability we calculate, we determine that the expected frequency is 120. This means, out of 720 tosses, you can anticipate that a sum greater than 9 will appear 120 times on average. It's important to note that expected frequency doesn't guarantee precise results for every set of trials, but rather serves as a statistical average over many observations.

Dice Probability

Dice Probability involves understanding the likelihood of various outcomes when rolling dice. With a standard six-sided die, each face holds an equal chance of appearing, leading to the simple calculation of probability. For our example of two dice, consider these points:

• There are 6 faces per die, leading to 36 total possible outcomes since every face of one die can pair with every face of the other die (6 x 6).
• Each face or combination, such as rolling a 5 and a 4, represents an equal likelihood.

Dice probability is foundational in games involving dice, enabling you to calculate the chances of different roll outcomes. For a more complex event like achieving a sum greater than 9, you first identify the specific outcomes leading to that sum, which we found to be 6 favorable results out of the 36 possibilities.

Probability Calculation

To understand probability calculation, it’s essential to see it as a ratio of favorable outcomes to possible outcomes. This calculation forms the core of determining probability across various scenarios, like our dice example.Here's how to perform a probability calculation:

• Count the total number of successful or favorable outcomes.
• Divide this number by the total number of possible outcomes.

In our dice scenario, there are 6 favorable outcomes where the sum of two dice is greater than 9 (combinations like (4,6), (6,6), etc.). With 36 total possibilities, the probability calculation is straightforward: \[\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{6}{36} = \frac{1}{6}.\]This probability helps determine how likely it is to roll a sum over 9 in any single toss of two dice.

Outcomes Analysis

Outcomes Analysis involves listing and evaluating all possible results from an experiment, especially in games involving chance, like rolling dice. It's about ensuring we consider every possible result and identify the ones that fit our criteria for success. The process includes:

• Identifying all potential outcomes, like the 36 forms when rolling two dice.
• Determining which outcomes meet the event criteria, such as having a sum greater than 9.

Through outcomes analysis, you ensure precise and comprehensive calculations. For our exercise, we meticulously list all outcomes yielding a sum over 9, which are (4,6), (5,5), (6,4) for a sum of 10; (5,6), (6,5) for a sum of 11; and (6,6) for a sum of 12, totaling 6 possible outcomes. This accuracy directly feeds the probability and expected frequency calculations, ensuring they are correct and reliable.