Jonah Patel
Probability is a useful concept that helps us understand the likelihood of an event occurring. It is often used in everyday situations, such as tossing a coin, rolling dice, or predicting the weather. The probability of an event can be calculated by dividing the number of favourable outcomes by the total number of possible outcomes. This results in a value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. For example, the probability of rolling a 6 on a single die is 1/6, while the probability of rolling an even number is higher at 1/2. When all outcomes are equally likely, probability calculations become more straightforward. However, in real-world situations, subjective probabilities come into play, influenced by individual beliefs and experiences. Understanding probability helps us make informed decisions and predictions about the world around us.
| Characteristics | Values |
|---|---|
| Probability of an event | 0 to 1 |
| Probability of an impossible event | 0 |
| Probability of a certain event | 1 |
| Probability formula | Favourable outcomes / Total outcomes = x/n |
| Probability of drawing a black ball from a bag containing red, blue, green, and yellow balls | 0 |
| Probability of drawing Ⓡ | 2/5 |
| Probability of drawing Ⓡ or Ⓑ | 3/5 |
| Probability of a coin landing on heads or tails | 1/2 |
| Probability of a coin landing on heads, tails, or on edge | Not 1/3, as the probability of landing on its edge is minuscule |
| Probability of getting a sum of 9 when throwing two dice | 4/36 or 1/9 |
| Probability of the Dow Jones average going up tomorrow | Subjective |
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What You'll Learn
Probability of an event
Probability is a useful concept that helps us understand the likelihood of an event occurring. It is often used in real-life situations, such as rolling a die, predicting the weather, or calculating the chances of winning or losing in sports.
At its core, probability is calculated by dividing the favourable number of outcomes by the total number of possible outcomes. This value always falls between 0 and 1, as the favourable outcomes can never exceed the total outcomes. For example, when tossing a coin, there are only two possible outcomes: heads or tails. The probability of each outcome is 1/2, or 0.5.
The probability of an event that is certain to occur is always 1. For instance, when rolling a die, it is certain that one of the six faces will be shown, so the probability of this event is 1. Similarly, the probability of an impossible event is always 0. If a bag contains only red, blue, green, and yellow balls, the probability of drawing a black ball is 0.
Subjective probability is an interesting perspective that considers an individual's belief that an event will occur. For example, in a legal context, a legal expert might assess the probability of different settlement outcomes based on their knowledge of similar cases and the current trends in the legal system. However, subjective probability can vary between individuals, and it must follow certain "coherence" conditions to be workable.
Probability theory also includes terms like "experiment," referring to a trial or operation conducted to produce an outcome, and "sample space," which encompasses all the possible outcomes of an experiment. Understanding these concepts helps provide a more comprehensive understanding of probability and its applications.
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Theoretical probability
Probability is the likelihood of an event occurring. Theoretical probability is a way of calculating this chance without conducting any experiments. It assumes that all outcomes are equally likely to occur.
The formula for theoretical probability is:
> Theoretical Probability = Number of favorable outcomes / Number of possible outcomes
For example, if there are 5 cards and you want to determine the probability of drawing 2 cards, the number of favourable outcomes (2) is divided by the total possible outcomes (5) to get the probability as 0.4.
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Conditional probability
For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing. So, the conditional probability that someone who is sick is coughing might be 75%. Here, the probability of the event "coughing" changes based on the occurrence of another event, "being sick".
Calculating conditional probability involves multiplying the probability of the preceding event by the updated probability of the succeeding, or conditional, event. This can be expressed mathematically as:
> P(A|B) = P(B|A) * P(A)
Where P(A|B) is the conditional probability of event A given event B, and P(B|A) is the probability of event B given event A.
It's important to note that conditional probability is different from independent probability, where the occurrence of one event does not affect the probability of another event. In conditional probability, the events are considered dependent, meaning that the occurrence of one event influences the likelihood of the other event occurring.
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Subjective probability
In contrast to subjective probability, there are two other main types of probabilities: empirical probability and classical probability. Empirical probability refers to the probability based on historical data, while classical probability is determined by formal reasoning and mathematical mechanisms. For example, the empirical probability of getting a head in a coin toss, given three previous tosses that resulted in heads, would be 100%. However, the classical or theoretical probability of getting a head in a coin toss is always 50%, as the outcome is binary and equally likely.
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Classical, empirical, and axiomatic perspectives
There are four main types of probability: classical, empirical, subjective, and axiomatic.
Classical Perspective
The classical perspective on probability, also known as theoretical probability, assumes that all outcomes of a given event are equally likely. It is calculated by defining an event and then determining the probability of that event as a ratio of the number of successful outcomes to the total number of possible outcomes. For example, if we toss a coin once and get the side we wanted, the formula would read P(S) = 1/2. This perspective is conceptually simple for a lot of situations, however, it has limits as many situations don't have a finite number of equally likely outcomes.
Empirical Perspective
Empirical probability, also known as experimental probability, evaluates outcomes based on conducting experiments. It uses the number of occurrences of a given outcome within a sample set as a basis for determining the probability of that outcome occurring again. For instance, if you roll a weighted die a number of times without knowing the weighted side, you can determine the proportion of times the die produces a desired outcome, and that outcome will then be the probability. The empirical perspective is used in most statistical inference procedures, however, it leaves open the question of how large the sample size has to be before we get a good approximation.
Axiomatic Perspective
The axiomatic perspective is a unifying theory of probability that sets out a series of rules that apply to all types of probability calculations, based on Kolmogorov's Three Axioms. The axiomatic perspective codifies these coherence conditions, so it can be used with any of the above three perspectives. The three axioms of probability are:
- 0 ≤ P(E) ≤ 1 for every allowable event E (in other words, 0 is the smallest allowable probability and 1 is the largest allowable probability).
- The certain event has probability 1 (for example, in rolling a die, the certain event is "one of 1, 2, 3, 4, 5, or 6 comes up").
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Frequently asked questions
1.
Only 1.
B) All events are equally likely in any probability procedure.
1. -.41, 1.22, √2 and 5/3. This is because the probability of an event happening can lie between 0 and 1.
