Chapter 4: Probability – Basic Concepts and Approaches

Probability

Basic Concepts of Probability

Probability is a fundamental concept in statistics that quantifies the likelihood of events occurring. It is expressed as a value between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event. Understanding probability allows us to interpret statistical results and make informed decisions based on data.

• Event: Any collection of results or outcomes from 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. The sample space lists all possible outcomes.

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}

Notation for Probabilities

Probabilities are denoted by P. For example, P(A) represents the probability of event A occurring. Events are often labeled as A, B, or C. Probabilities can be expressed as fractions, decimals, or rounded to three significant digits.

• P(A): Probability of event A occurring.

• P(not A): Probability that event A does not occur.

Three Common Approaches to Finding the Probability of an Event

There are three main approaches to determining the probability of an event:

1. Relative Frequency Approximation of Probability: Probability is estimated by conducting or observing a procedure and counting the number of times event A occurs.

$P(A) = \frac{\text{number of times A occurred}}{\text{number of times the procedure repeated}}$

2. Classical Approach to Probability (Requires Equally Likely Outcomes): If a procedure has n different simple events that are equally likely, and event A can occur in s different ways:

$P(A) = \frac{\text{number of ways A occurs}}{\text{number of different simple events}}$

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

• Simulations: When none of the above approaches are possible, simulations can be used to model the procedure and estimate probabilities.

Law of Large Numbers

The law of large numbers states that as a procedure is repeated many times, the relative frequency probability of an event tends to approach the actual probability. This law applies to behavior over a large number of trials, not to individual outcomes.

Complementary Events

The complement of event A, denoted by A', consists of all outcomes in which event A does not occur. The probability of the complement is:

$P(A') = 1 - P(A)$

• Example: If the probability of dying in a skydiving jump is 0.000007, then the probability of surviving is $1 - 0.000007 = 0.999993$.

Identifying Significant Results with Probabilities

Probabilities can be used to determine when results are significantly high or low using the rare event rule for inferential statistics:

• Significantly High Number of Successes: x successes among n trials is significantly high if $P(x \text{ or more}) < 0.05$.

• Significantly Low Number of Successes: x successes among n trials is significantly low if $P(x \text{ or fewer}) < 0.05$.

Note: The value 0.05 is a common threshold but not absolutely rigid.

Odds

Odds are another way to express the likelihood of events, commonly used in gambling and lotteries.

• Actual Odds Against Event A: The ratio $P(A') : P(A)$, usually expressed as "a to b" where a and b are integers.

• Actual Odds in Favor of Event A: The reciprocal ratio $P(A) : P(A')$.

• Payoff Odds Against Event A: The ratio of net profit (if you win) to the amount bet.

Example: In roulette, betting $5 on the number 13 (probability of winning is $1/38$):

• Actual odds against 13: $\frac{P(\text{not 13})}{P(13)} = \frac{37/38}{1/38} = 37:1$

• Casino payoff odds: 35:1 (net profit of $35 for each $1 bet)

• For a $5 bet, net profit is $5 \times 35 = $175

• If the casino used actual odds (37:1), net profit would be $5 \times 37 = $185

Summary Table: Key Terms and Formulas

Key Terms	Key Formulas/Algorithms
Event, Sample Space, Law of Large Numbers, Rare Event Rule, Complementary Events	Relative Frequency Approximation: $P(A) = \frac{\text{number of times A occurred}}{\text{number of times the procedure repeated}}$ Classical Approach: $P(A) = \frac{\text{number of ways A occurs}}{\text{number of different simple events}}$ Odds For, Odds Against, Payoff Odds

Conclusion

• Probability values and odds are essential for interpreting statistical results.

• Probabilities help determine when results are significantly high or low.

• Understanding and computing probabilities and odds are foundational skills in statistics.