4.1: Probability, Sample Spaces, and Probability models Flashcards
What is the definition of probability?
Probability is a number that measures the chance or likelihood that an event will occur.
It is always a value between 0 and 1, where 0 indicates the event will not occur, and 1 indicates certainty that the event will occur.
How is the likelihood of an event related to its probability value?
The likelihood of an event is directly related to its probability value.
The closer the probability is to 1, the higher the likelihood of the event occurring.
Conversely, the closer the probability is to 0, the smaller the likelihood of the event occurring.
What is an experiment in the context of statistical studies?
4: An experiment refers to the method of data collection in statistical studies.
It can involve either performing a controlled experiment, where specific conditions are varied, or observing uncontrolled events, such as recording stock prices over a period.
How is the sample space of an experiment defined?
The sample space of an experiment is the set of all possible outcomes or results that can occur. In the context of statistical studies, it encompasses all the potential outcomes that can be observed or measured during the experiment.
What is an experiment in the context of probability theory?
An experiment in probability theory is any process of observation that has an uncertain outcome.
It can involve activities like tossing a coin, rolling a die, or conducting scientific tests, where the result is uncertain.
The sample space of an experiment is the set of all possible outcomes for that experiment.
Sample space outcomes are the specific results that can occur in a single repetition of the experiment.
These outcomes must be clearly defined so that only one outcome can occur in a single repetition.
For example, when tossing a coin, the sample space outcomes are “head” and “tail,” and when rolling a die, the outcomes are numbers 1 to 6.
Why is it necessary to define sample space outcomes clearly?
Defining sample space outcomes clearly is essential because it ensures that, in any single repetition of the experiment, one and only one sample space outcome will occur.
This precision is vital for accurate probability calculations and understanding the likelihood of different outcomes.
What are the three methods commonly used to assign probabilities to sample space outcomes?
The three methods used to assign probabilities to sample space outcomes are the classical method,
the relative frequency method,
and the subjective method.
What are the two conditions that must be met when assigning probabilities to sample space outcomes?
When assigning probabilities to sample space outcomes, two conditions must be met:
The probability assigned to each sample space outcome must be between 0 and 1, denoted as
0≤P(E)≤1, where E represents a sample space outcome.
The probabilities of all sample space outcomes must sum to 1.
How is the classical method of assigning probabilities applied, and when is it appropriate to use this method?
The classical method is used when sample space outcomes are equally likely. For example, in the experiment of tossing a fair coin (with outcomes “head” and “tail”), both outcomes are equally likely. In this method, the probability of each outcome is 1/N, where N is the number of equally likely outcomes.
For instance, when rolling a fair die (with outcomes 1, 2, 3, 4, 5, and 6), each outcome is assigned a probability of 1/6 because there are 6 equally likely outcomes.
Explain the logic behind assigning probabilities using the classical method.
In the classical method, when all sample space outcomes are equally likely, the logic dictates that the probability of each outcome is 1/N, where N is the number of equally likely outcomes.
This ensures that each outcome has a probability between 0 and 1, and the sum of probabilities for all outcomes equals 1.
For example, in tossing a fair coin, both “head” and “tail” are equally likely, so each has a probability of 1/2, which satisfies the conditions of probabilities being between 0 and 1 and summing up to 1.
What is the long-run relative frequency interpretation of probability?
The long-run relative frequency interpretation of probability states that probability is interpreted as a long-run relative frequency.
In other words, if an experiment is repeated a large number of times (approaching infinity), the probability of an event is the proportion of times that event occurs in these repetitions.
For example, if a fair coin is tossed many times, the probability of getting heads (.5) means that in the long run, heads will appear in 50 percent of the tosses.
How is the relative frequency of an event calculated, and what does it represent?
The relative frequency of an event is calculated by dividing the number of times the event occurs by the total number of repetitions or trials.
It represents the fraction or proportion of times the event occurs in a finite set of trials.
For example, if a coin is tossed 10 times and 6 times it shows heads, the relative frequency of heads is 6/10=0.6.
Why is the relative frequency interpretation of probability considered a mathematical idealization?
The relative frequency interpretation of probability is a mathematical idealization because it assumes an infinite number of repetitions of an experiment, which is practically impossible.
While it provides a theoretical basis for understanding probabilities, it is not always feasible to perform experiments an indefinitely large number of times in reality.
How is probability often perceived in terms of the percentage of time an outcome would occur in many repetitions of the experiment?
Probability is often perceived as the percentage of time a sample space outcome would occur in many repetitions of the experiment.
For example, if the probability of obtaining a head when tossing a coin is 0.5, it means that in many repetitions of tossing the coin, a head would occur 50 percent of the time. This interpretation helps in understanding probabilities in practical terms.
