Using real-life scenarios involving drunk driving accident rates and cell phone usage, the concept of conditional probability, how to interpret conditional probabilities in real-life contexts, and how to calculate conditional probabilities by using the formula P(A and B)/P(B) is explained.
After completing this tutorial, you will be able to complete the following:
The probability of an event measures the event's likelihood, or chance, of occurring. Special vocabulary is used in order to precisely define this concept. Outcomes are the results of situations involving chance or uncertainty. Such situations are sometimes called experiments, but this word can be misleading. Examples of experiments include flipping a coin, measuring the height of a randomly chosen person, or measuring the magnitude of an earthquake.
A sample space is a collection of all outcomes under consideration. For example, the possible outcomes of rolling a six-sided die could be the six different ways for the die to land. In this case, the sample space is given by where each outcome corresponds to a face of the die. Subcollections of outcomes in sample spaces are called events. Given the definition of A above, the sets and are all events of U. In the context of rolling a die, these events correspond to rolling a number between 1 and 3, the number 1, and a number between 1 and 6, respectively.
The probability of an event is the ratio of the event's size to the size of the sample space. For example, the probability of rolling a three is equal to because there is one way out of six possible outcomes for this to occur. On the other hand, the probability of rolling a number between one and three is equal to or 0.5. In order to provide a general definition of probability, let be a nonempty sample space and some event in U. The probability of denoted by P(E), is given by the following formula, in which and denote the size of and respectively:
This machinery makes it relatively simple to find more complicated probabilities. For example, if measuring probabilities related to drawing one card from a standard deck of 52 cards, the sample space U could consist of all 52 cards in the deck.
To calculate the probability of drawing a queen, we first define the event.
Since and the probability is calculated as follows:
These conventions also make it relatively straightforward to discuss compound probabilities. For example, to find the probability of drawing a card that has a red suit and an even number, we define the following events:
The probability is denoted by P(E & R), where E & R denotes the outcomes contained in both E and R. It is straightforward to verify that there are 10 such cards.
The probability above is an example of a conjunctive probability, which is the probability of two or more events occurring simultaneously. Conditional probabilities, which measure the probability of event given that another event happens, are another type of compound probability. Intuitively, conditional probabilities are used to capture the idea that the probabilities of some events depend on other events. For example, the probability of a houseplant dying can depend on whether or not the plant has been watered.
Given two events, A and B with B nonempty, the probability of A given B is denoted by and defined as follows.
To understand conditional probabilities, consider the following two questions related to drawing a single card from a standard deck.
What is the....
The first question can be answered by using the standard technique of defining the event and calculating quotients. We have and therefore
The second question requires a subtle reading in order to accurately provide its answer. To see this, imagine a card guessing game in which one player draws a card, tells the other player some information about the card, and the other player is expected to guess the card. It would be foolish to guess three of hearts if told that the card is an even number. On the other hand, guessing two would be reasonable if told the card is an even number. Other examples of conditional probability include the probability of having a car accident if alcohol-impaired or the probability of being an Internet-user if living in China.
In certain probability questions, the size of the underlying events cannot be determined, such as when inferring probabilities from well-known statistics, or calculating probabilities on continuous spaces. In these cases, a formula in terms of probabilities is required. Such a formula is derived below from counting-based formula by using the definition of and
The above derivation does not work for continuous probabilities; in such cases, a more advanced probability theory is required. Many texts actually use the formula derived above to define conditional probability. This has the drawback of slightly obscuring the intuitive meaning of the concept.
|Approximate Time||30 Minutes|
|Pre-requisite Concepts||Students should know the concepts of probability, compound probability, and event; and be able to perform elementary probability calculations.|
|Type of Tutorial||Concept Development|
|Key Vocabulary||conditional probability, data, event|