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Chapter 5: Problem 4

A person was trying to figure out the probability of getting two heads when flipping two coins. He flipped two coins 10 times, and in 2 of these 10 times. both coins landed heads. On the basis of this outcome, he claims that the probability of two heads is \(2 / 10\), or \(20 \%\). Is this an example of an empirical probability or a theoretical probability? Explain.

Short Answer

Expert verified
This is an example of an empirical probability because the person in the exercise based the probability on observable results of the experiment.

Step by step solution

01

Understand Different Types of Probability

Empirical probability is determined by carrying out a series of experiments and calculating the ratio of the favorable outcome to the total number of trials. Theoretical probability, on the other hand, is found by assuming that all outcomes are equally likely and then calculating the ratio of favorable outcomes to the total number of outcomes.
02

Identify Type of Probability in Exercise

The person in the exercise made an observation by flipping coins a number of times and recorded the outcomes. He then concluded a probability from those outcomes. This is a classic case of empirical probability because it is based on observations.
03

Validate the Answer

The statement 'On the basis of this outcome, he claims that the probability of two heads is \(2 / 10\), or \(20 \%\).' solidifies our conclusion that this is an empirical probability because he based his probability on experimental results.

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Key Concepts

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

Types of Probability
Probability is a fascinating concept in mathematics that helps us understand the likelihood of events occurring. There are two main types that students often encounter: empirical probability and theoretical probability.
  • Empirical Probability: This type is derived from actual experiments or observations. For example, if you flip a coin multiple times and record the outcomes, the probability you calculate from this data is empirical. It represents the observed frequency of the outcome happening.

  • Theoretical Probability: In contrast, theoretical probability is calculated based on the assumption that all outcomes are equally likely, without conducting any actual experiments. It involves mathematical reasoning and logical deduction.
Understanding the difference between these two types is crucial for interpreting probability exercises correctly.
Theoretical Probability
Theoretical probability is all about prediction using logical and mathematical tools. For events where outcomes are equally likely, theoretical probability can be easily determined.
  • If you're rolling a fair six-sided die, each side (number) has an equal chance of landing face up. Since there are six sides, the theoretical probability of rolling a specific number, like a 3, is \(1/6\).

  • Theoretical probability relies on mathematical formulas. For a two-coin flip, the total possible outcomes are four: HH (two heads), HT (a head and a tail), TH (a tail and a head), TT (two tails). Thus, to get two heads, the theoretical probability is \(1/4\) or 25%.
It鈥檚 important to note that theoretical probability is not hindered by physical constraints or real-world variations. It assumes an ideal situation that might never physically occur.
Experimental Results
Experimental results lay the foundation for empirical probability. Here, probability is determined through actual trials and observations of real-life phenomena.
  • In the coin flip example, the person flipped two coins ten times and noted the times both coins landed on heads. The experimental result showed this occurred twice.

  • The result gives us empirical insights: \(\frac{2}{10}\) or 20% probability of getting two heads based on what was experimentally observed during the trials.

  • Unlike theoretical probability, experimental results may vary with each set of trials. This variability is due to chance, experimental errors, or differences in the conditions of the experiment.
Conducting multiple sets of experiments and pooling results can often give a more accurate representation.
Probability Calculation
Calculating probability necessitates understanding both theoretical and empirical methods. Here鈥檚 a step-by-step for how you would calculate probability in both contexts:
  • For Empirical Probability: Use the formula \(\text{Empirical Probability} = \frac{\text{Number of times event occurs}}{\text{Total number of trials}}\) to find the probability from experimental data.

  • For Theoretical Probability: Identify all possible outcomes first. Then, use the formula \(\text{Theoretical Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\).

By comparing the probabilities calculated through each method, we can determine how theoretical assumptions fare against practical evidence. This process is invaluable in fields ranging from science to finance, where understanding the likelihood of different outcomes is essential.

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Most popular questions from this chapter

Suppose a person is selected at random from a large population. a. Label each pair of events as mutually exclusive or not mutually exclusive. i. The person has traveled to Mexico; the person has traveled to Canada. ii. The person is single; the person is married. b. Give an example of two events that are mutually exclusive when a person is selected at random from a large population.

A multiple-choice test has 30 questions. Fach question has three choices, but only one choice is correct. Using a random number table, which of the following methods is a valid simulation of a student who circles his or her choices randomly? Explain. (Note: there might be more than one valid method.) a. The digits 1,2 , and 3 represent the student's attempt on one question. All other digits are ignored. The 1 represents the correct choice, the 2 und 3 represent incorrect choices. b. The digits \(0,1,4\) represent the student's attempt on one question. All other digits are ignored. The 0 represents the correct choice, the 1 and 4 represent incorrect choices. c. Lach of the 10 digits represents the student's attempt on one question. The digits \(1,2,3\) represent a correct choice; \(4,5,6,7,8,9\) and 0 represent an incorrect choice.

When a certain type of thumbtack is tossed, the probability that it lands tip up is \(60 \%\), and the probability that it lands tip down is \(40 \%\). All possible outcomes when two thumbtacks are tossed are listed. U means the tip is Up, and D means the tip is Down. \(\begin{array}{llll}\text { UU } & \text { UD } & \text { DU } & \text { DD }\end{array}\) a. What is the probability of getting exactly one Down? b. What is the probability of getting two Downs? c. What is the probability of getting at least one Down (one or more Downs)? d. What is the probability of getting at most one Down (one or fewer Downs)?

Use your general knowledge to label the following pairs of variables as independent or associated. Fxplain. a. The outcome on flips of two separate, fair coins b. Breed of dog and weight of dog for dogs at a dog show

Assume that the only grades possible in a history course are \(\mathrm{A}, \mathrm{B}, \mathrm{C}\), and lower than \(\mathrm{C}\). The probability that a randomly selected student will get an \(\mathrm{A}\) in a certain history course is \(0.18\), the probability that a student will get a \(\mathrm{B}\) in the course is \(0.25\), and the probability that a student will get a \(\mathrm{C}\) in the course is \(0.37\). a. What is the probability that a student will get an A OR a B? b. What is the probability that a student will get an A OR a B OR a C? c. What is the probability that a student will get a grade lower than a \(\mathrm{C}\) ?
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