Introduction

Chance is an everyday part of our lives. Although many outcomes are determined by chance, a pattern can be observed from many repetitions. Mathematics can be used to understand the regular patterns of chance behaviour when the same chance process is repeated multiple times and this is called probability.

• Chance is the possibility of something happening.

Two-way tables, Venn diagrams, and tree diagrams are some helpful tools for displaying possible outcomes from a chance process.

Simulation is a very useful method to model chance behaviour.

E.g. In a convenience store, every 20-ounce bottle of soda for sale has a 1-in-6 chance of getting a cap that says, “You’re a winner!” The status of an individual bottle can be modelled with a six-sided die.

Probability calculations are useful because they are the basis for inference.

The rules of probability provide an answer to the question "What would happen if we repeated the random sampling or random assignment procedure many times?" when we create data through randomised comparison studies or sampling at random.

5.1- Randomness, Probability, and Simulation

A better understanding of how chance behaviour operates can answer many questions in everyday life such as predicting the outcomes of tossing a coin 10 times.

The Idea of Probability

The basis idea for probability is that chance behaviour is unpredictable in the short run but has a regular and predictable pattern in the long run.

If we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value. This value is called probability.

• The probability.of any outcome of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions.

Probability gives us a method to describe the long-term regularity of random behaviour. Outcomes that never occur have probability 0. An outcome that happens on every repetition has probability 1. It is not possible to observe a probability exactly because a coin can always be tossed continuously.

Myths about Randomness

One myth about randomness is that future outcomes must make up for an imbalance. Actually, the law of large numbers in action usually overwhelms any discrepancy or unbalance over the long run.

E.g. When tossing a coin six times and getting TTTTTT, the next toss must be more likely to give a head. Actually after 10,000 tosses, results of the first six tosses don't matter - it's the next 9994 that matter.

Simulation

Simulations follow a basic strategy which can be summarized by a four-step process: State, Plan, Do, Conclude.

• The imitation of chance behaviour, based on a model that accurately reflects the situation, is called a simulation.

**State:**Ask a question of interest about some chance process.
Plan: Describe how to use a chance device to imitate one repetition of the process. Tell what you will record at the end of each repetition.

Do: Perform many repetitions of the simulation.

Conclude: Use the results of your simulation to answer the question of interest.

Besides physical devices, technology can also be used to create simulations.

Simulations only suggest the likelihood of chance events happening but not actuality.

5.2- Probability Rules

Chance behaviour is predictable in the long run.

It is not necessary to always use simulations to imitate chance behaviour in order to determine the probability of a particular outcome.

Probability Models

Chance behaviour can be modelled instead of always using simulations.

A probability model does more than just assign a probability to each outcome. It

allows us to find the probability of any collection of outcomes, which we call an event.

• Event is any collection of outcomes from some chance process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on.

A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome.

The sample space S of a chance process is the set of all possible outcomes.

Basic Rules of Probability

• The probability of any event is a number between 0 and 1.

• All possible outcomes together must have probabilities that add up to 1.

• If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula

  P(A) = number of outcomes corresponding to event A/ total number of outcomes in sample space

• The probability that an event does not occur is 1 minus the probability that the event does occur.

• If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.

When two events have no outcomes in common, we refer to them as mutually exclusive mutually exclusive or disjoint.

• Two events A and B are mutually exclusive (disjoint) if they have no outcomes in common and so can never occur together—that is, if P (A and B ) = 0.

Two-Way Tables, Probability, and the General Addition Rule

A two-way table can make probability calculations easier because it displays sample space in a clearer way.

General addition rule: If A and B are any two events resulting from some chance process, then P(A or B) = P(A) + P(B) − P(A and B)

Venn Diagrams and Probability

Venn diagrams can also be used to show the sample space of a chance process with two events. Important concepts include complement, intersection, and union.

• Complement: The probability that an event does not occur is 1 minus the probability that the event does occur. In symbols, P(AC) = 1 – P(A).

• Intersection: The intersection of events A and B, denoted by A n B, refers to the occurrence of both of two events at the same time.

• Union The union of events A and B, denoted by A U B, consists of all outcomes in A or B or both. (p. 311)

5.3- Conditional Probability

The probability of an event can change if there is a known occurrence of another event.

This concept is useful in calculating probability.

What Is Conditional Probability?

Conditional probability is the probability that one event will happen under the condition that some other event is already known to have occurred. E.g. The probability that a randomly selected Harvard student reads the Wall Street Journal also reads the New Yorker.

Conditional probability formula: p(A|B) = P(A∩B) / P(B)

p(A|B) is the probability of event A occurring, given that event B occurs.

The General Multiplication Rule and Tree Diagrams

The probability that events A and B both occur can be found using the general multiplication rule: P(A and B) = P(A ∩ B) = P(A) · P(B | A)

where P(B | A) is the conditional probability that event B occurs given that event A has already occurred. E.g

This rule states that the first event must first occur, and then the second event must occur as a result of the first occurrence. Very useful when a chance process involves a sequence of outcomes.

Tree diagram can be used to model the general multiplication rule when a chance process involves a sequence of outcomes.

Conditional Probability and Independence

Two events are independent when the occurrence of one event does not affect the probability of the other. In other words, events A and B are independent if P(A | B) = P(A) and P(B | A) = P(B).

E.g Tossing a coin twice.

Independence: A Special Multiplication Rule

When two events are independent, the general multiplication rule does not apply. Another rule called the special multiplication rule for independent events apply.

If A and B are independent events, then the probability that A and B both occur is P (A ∩ B ) = P (A) . P (B )

Two mutually exclusive events can never be independent, because if one event happens, the other event is guaranteed not to happen. E.g. Chance someone is a male and chance someone is pregnant.

The addition rule P(A or B) = P(A) + P(B) holds if A and B are mutually exclusive but not otherwise.