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Chapter 5: Problem 4

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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Most popular questions from this chapter

The Wall Street Journal regularly publishes an article entitled "The Count." In one article, The Count looked at 1000 randomly selected home runs in Major League Baseball. (a) Of the 1000 homeruns, it was found that 85 were caught by fans. What is the probability that a randomly selected homerun is caught by a fan? (b) Of the 1000 homeruns, it was found that 296 were dropped when a fan had a legitimate play on the ball. What is the probability that a randomly selected homerun is dropped? (c) Of the 85 caught balls, it was determined that 34 were barehanded catches, 49 were caught with a glove, and two were caught in a hat. What is the probability a randomly selected caught ball was caught in a hat? Interpret this probability. (d) Of the 296 dropped balls, it was determined that 234 were barehanded attempts, 54 were dropped with a glove, and eight were dropped with a failed hat attempt. What is the probability a randomly selected dropped ball was a failed hat attempt? Interpret this probability.

Shawn is planning for college and takes the SAT test. While registering for the test, he is allowed to select three schools to which his scores will be sent at no cost. If there are 12 colleges he is considering, how many different ways could he fill out the score report form?

Fingerprints are now widely accepted as a form of identification. In fact, many computers today use fingerprint identification to link the owner to the computer. In \(1892,\) Sir Francis Galton explored the use of fingerprints to uniquely identify an individual. A fingerprint consists of ridgelines. Based on empirical evidence, Galton estimated the probability that a square consisting of six ridgelines that covered a fingerprint could be filled in accurately by an experienced fingerprint analyst as \(\frac{1}{2}\). (a) Assuming that a full fingerprint consists of 24 of these squares, what is the probability that all 24 squares could be filled in correctly, assuming that success or failure in filling in one square is independent of success or failure in filling in any other square within the region? (This value represents the probability that two individuals would share the same ridgeline features within the 24 -square region.) (b) Galton further estimated that the likelihood of determining the fingerprint type (e.g., arch, left loop, whorl, etc.) as \(\left(\frac{1}{2}\right)^{4}\) and the likelihood of the occurrence of the correct number of ridges entering and exiting each of the 24 regions as \(\left(\frac{1}{2}\right)^{8}\). Assuming that all three probabilities are independent, compute Galton's estimate of the probability that a particular fingerprint configuration would occur in nature (that is, the probability that a fingerprint match occurs by chance).

Determine the probability that at least 2 people in a room of 10 people share the same birthday, ignoring leap years and assuming each birthday is equally likely, by answering the following questions: (a) Compute the probability that 10 people have 10 different birthdays. Hint: The first person's birthday can occur 365 ways, the second person's birthday can occur 364 ways, because he or she cannot have the same birthday as the first person, the third person's birthday can occur 363 ways, because he or she cannot have the same birthday as the first or second person, and so on. (b) The complement of "10 people have different birthdays" is "at least 2 share a birthday." Use this information to compute the probability that at least 2 people out of 10 share the same birthday.

Why is the following not a probability model? $$ \begin{array}{lc} \text { Color } & \text { Probability } \\ \hline \text { Red } & 0.1 \\ \hline \text { Green } & 0.1 \\ \hline \text { Blue } & 0.1 \\ \hline \text { Brown } & 0.4 \\ \hline \text { Yellow } & 0.2 \\ \hline \text { Orange } & 0.3 \end{array} $$
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