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Suppose that you roll a pair of dice 1000 times and get seven 350 times. Based on these results, what is the probability that the next roll results in seven?

Short Answer

Expert verified
The probability is 0.35 or 35%.

Step by step solution

01

Understanding Probability

Probability is the measure of the likelihood of an event occurring. It is calculated by dividing the number of favorable outcomes by the total number of trials.
02

Identifying Favorable Outcomes

From the problem, the number of times a seven has been rolled (favorable outcomes) is 350.
03

Identifying Total Trials

The total number of trials, which is the total number of dice rolls, is 1000.
04

Calculate Experimental Probability

Use the formula for probability: \[ P(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Trials}} \]Substitute the values to get: \[ P(7) = \frac{350}{1000} = 0.35 \]
05

Interpretation

The probability that the next roll results in seven, based on the experimental data, is 0.35 or 35%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is a concept in mathematics that measures the chance or likelihood of a particular event happening. It's usually expressed as a ratio or a fraction.
If we dice roll, rather than listing multiple outcomes, we calculate probability using a simple formula. The formula for probability is:
\[ P(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Trials}} \]
The value of probability ranges from 0 to 1. A probability of 0 indicates that an event will not occur, and a probability of 1 indicates that an event certainly will occur. Any fraction or decimal between these numbers represents the likelihood of an event based on past data or theoretical outcomes.
Favorable Outcomes
To understand probability better, let's dive into the idea of favorable outcomes.
Favorable outcomes are the specific results of an experiment that we are interested in. In our example, getting a seven when rolling a pair of dice is our favorable outcome.
Understanding these outcomes is essential because they form the numerator in our probability formula.
In the given exercise, rolling a seven occurred 350 times during the 1000 dice rolls. This tells us that the number of favorable outcomes is 350. For each experiment or trial, you should first clearly identify what you want to measure to move forward effectively.
Total Trials
Total trials refer to the total number of times an experiment is conducted. It's crucial because it forms the denominator in our probability formula.
In the example of the exercise, the total number of dice rolls is 1000. Each time you roll the dice, it counts as one trial. The total trials provide the sample size, which is essential for calculating accurate and reliable probability.
The larger the number of trials, the more reliable your experimental probability is likely to be. This is why 1000 trials usually give us a good approximation of the actual probability. By dividing the number of favorable outcomes by the total trials, you can determine the experimental probability, giving you a ratio to predict future events based on past data.

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