Chapter 4: Probability – Basic Concepts and Approaches

Probability

4-1 Basic Concepts of Probability

Probability is a foundational concept in statistics, quantifying the likelihood of events occurring in a random experiment. Probabilities are always expressed as values between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

• Event: Any collection of results or outcomes of a procedure.

• Simple Event: An outcome or event that cannot be further broken down into simpler components.

• Sample Space: The set of all possible simple events for a procedure.

Example: Simple Events and Sample Spaces

Consider the outcomes of births, where "b" denotes a baby boy and "g" denotes a baby girl.

Procedure	Example of Event	Sample Space: Complete List of Simple Events
Single birth	1 girl (simple event)	{b, g}
3 births	2 boys and 1 girl (bbg, bgb, gbb are all simple events resulting in 2 boys and 1 girl)	{bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}

• Not a Simple Event: The event "2 girls and 1 boy" is not a simple event because it can occur in multiple ways (ggb, gbg, bgg).

Three Common Approaches to Finding the Probability of an Event

There are three main approaches to determining probabilities:

1. Relative Frequency Approximation: Probability is approximated by conducting (or observing) a procedure and counting the number of times event A occurs: $P(A) = \frac{\text{number of times A occurred}}{\text{number of times the procedure was repeated}}$

2. Classical Approach (Equally Likely Outcomes): If a procedure has n equally likely simple events and event A can occur in s ways: $P(A) = \frac{\text{number of ways A occurs}}{\text{number of different simple events}} = \frac{s}{n}$ Caution: Only use this approach if outcomes are equally likely.

3. Subjective Probability: Probability is estimated using knowledge of relevant circumstances when empirical or classical methods are not feasible.

Simulations

When none of the above approaches are practical, a simulation can be used. A simulation is a process that mimics the behavior of a real procedure to estimate probabilities.

Rounding Probabilities

• Express probabilities as exact fractions or decimals, or round to three significant digits for clarity.

• For non-simple fractions (e.g., $\frac{2}{3}$ or $\frac{5}{9}$), use decimals for better understanding.

Law of Large Numbers

As a procedure is repeated many times, the relative frequency probability of an event approaches the actual probability.

• The law applies to large numbers of trials, not individual outcomes.

• Do not assume outcomes are equally likely without evidence.

Examples

• Airline Crashes: Probability of a crash on a given flight: $P(\text{airline crash}) = \frac{16}{39,000,000} = 0.000000410$

• Ghosts Survey: Probability a randomly selected adult reports seeing a ghost: $P(\text{response of "yes"}) = \frac{366}{2003} = 0.183$

Complementary Events

• The complement of event A (denoted $\overline{A}$) consists of all outcomes where A does not occur.

• $P(A) + P(\overline{A}) = 1$

Example: Internet Users

• Probability a randomly selected adult does not use the Internet: $P(\text{not using the Internet}) = \frac{220}{2002} = 0.110$

Identifying Significant Results with Probabilities

• Rare Event Rule: If an observed event is very unlikely under a given assumption, and it occurs, the assumption is probably incorrect.

• Significantly High Number of Successes: $x$ is significantly high if $P(x \text{ or more}) \leq 0.05$

• Significantly Low Number of Successes: $x$ is significantly low if $P(x \text{ or fewer}) \leq 0.05$

Probability Review

• Probability values range from 0 (impossible) to 1 (certain).

• Notation: $P(A)$ is the probability of event A; $P(\overline{A})$ is the probability that event A does not occur.

Odds

• Actual Odds Against: $\frac{P(\overline{A})}{P(A)}$, usually expressed as a:b.

• Actual Odds in Favor: $\frac{P(A)}{P(\overline{A})}$, the reciprocal of the odds against.

• Payoff Odds: The ratio of net profit to the amount bet: Payoff odds against event A = (net profit):(amount bet)

Example: If the probability of winning a bet is $\frac{1}{38}$ and the payoff odds are 35:1, the actual odds against are 37:1, and the net profit for a $5 bet would be $175 (if the casino were fair, it would be $185).

Additional info: These notes cover the foundational probability concepts essential for further study in statistics, including probability rules, event classification, and practical calculation methods.