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Chapter 4: Problem 4

What is the law of large numbers? If you were using the relative frequency of an event to estimate the probability of the event, would it be better to use 100 trials or 500 trials? Explain.

Short Answer

Expert verified
The Law of Large Numbers suggests using 500 trials for a more accurate probability estimate.

Step by step solution

01

Understanding the Law of Large Numbers

The Law of Large Numbers is an important principle in probability and statistics. It states that as the number of trials increases, the average of the results obtained from these trials will get closer to the expected value or theoretical probability of the event. Essentially, the more you repeat an experiment, the more accurate your estimate of the probability will become.
02

Defining Relative Frequency

Relative frequency is defined as the number of times an event occurs divided by the total number of trials. It is used as an estimate of the actual probability of an event. With more trials, the relative frequency is expected to converge to the true probability of the event.
03

Comparing 100 Trials and 500 Trials

According to the Law of Large Numbers, using more trials will typically provide a better estimate of the true probability. With 100 trials, your approximation of the probability might be close, but it could still be significantly affected by random variations. With 500 trials, the relative frequency should be even closer to the true probability because the effect of random fluctuations is minimized.
04

Conclusion on Trials Needed

Given the Law of Large Numbers, it would be better to use 500 trials rather than 100 trials when estimating the probability of an event through its relative frequency. The larger number of trials will lead to a more accurate and reliable estimate of the probability.

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Key Concepts

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

Probability Estimation
Estimating probability is like trying to predict the outcome of an event based on previous experiences or experiments. Imagine you're flipping a coin. Initially, you might not know how often it will land on heads versus tails. By flipping it a certain number of times, you can start to estimate these probabilities. The more times you flip it, the closer your estimate will get to the actual probability.
This is because, over a large number of trials, the true odds become clearer. Probability estimation relies on the results of these repeated trials in order to determine the likelihood of specific outcomes.
Relative Frequency
Relative frequency is a simple yet powerful concept in statistics. It's calculated by dividing the number of times an event occurs by the total number of trials. For instance, if you roll a die 100 times and get the number 6 forty times, the relative frequency for rolling a 6 would be calculated as 40/100 = 0.4.
The beauty of relative frequency lies in its ability to approximate the probability of an event. The more trials you conduct, the more accurately this relative frequency mirrors the actual probability of what you're observing.
  • Think of it as the event's track record based on the trials.
  • It's constantly updated and refined with each new trial conducted.
Statistical Trials
Statistical trials are individual attempts or experiments performed to observe and record outcomes. Each trial is an opportunity to learn more about the probability of an event.
For example, when you toss a coin, each flip is a separate trial. The importance of trials grows with the Law of Large Numbers. As the number of trials increases, the average of their outcomes gets closer to the expected probability.
The key takeaways are:
  • Statistical trials gather data to provide empirical evidence for estimating probabilities.
  • Larger numbers of trials improve certainty and accuracy in probability estimation.
Simply put, higher numbers of trials reduce the margin for error and random chance skewing results, reinforcing reliable probability predictions.

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Most popular questions from this chapter

Consider the following events for a college student selected at random: \(A=\) student is female \(B=\) student is majoring in business Translate each of the following phrases into symbols. (a) The probability the student is male or is majoring in business (b) The probability a female student is majoring in business (c) The probability a business major is female (d) The probability the student is female and is not majoring in business (e) The probability the student is female and is majoring in business

You need to know the number of different arrangements possible for five distinct letters. You decide to use the permutations rule, but your friend tells you to use \(5 !\). Who is correct? Explain.

M\&M plain candies come in various colors. According to the M\&M/Mars Department of Consumer Affairs (link to the Mars company web site from the Brase/Brase statistics site at college.hmco.com/pic/braseUS9e), the distribution of colors for plain M\&M candies is \(\begin{array}{l|ccccccc} \hline \text { Color } & \text { Purple } & \text { Yellow } & \text { Red } & \text { Orange } & \text { Green } & \text { Blue } & \text { Brown } \\ \hline \text { Percentage } & 20 \% & 20 \% & 20 \% & 10 \% & 10 \% & 10 \% & 10 \% \\ \hline \end{array}\) Suppose you have a large bag of plain M\&M candies and you choose one candy at random. Find (a) \(P(\) green candy or blue candy). Are these outcomes mutually exclusive? Why? (b) \(P(\) yellow candy or red candy). Are these outcomes mutually exclusive? Why? (c) \(P(\) not purple candy \()\)

If two events \(A\) and \(B\) are independent and you know that \(P(A)=0.3\), what is the value of \(P(A \mid B)\) ?

You toss a pair of dice. (a) Determine the number of possible pairs of outcomes. (Recall that there are six possible outcomes for each die.) (b) There are three even numbers on each die. How many outcomes are possible with even numbers appearing on each die? (c) Probability extension: What is the probability that both dice will show an even number?
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