Chapter 4: Probability – Fundamental Concepts and Applications

Probability

Basic Concepts of Probability

Probability is a measure of how likely an event is to occur, expressed as a value between 0 and 1. Understanding probability is essential for interpreting statistical results and making informed decisions under uncertainty.

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

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

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

Example: For a single birth, the possible simple events are 'boy' (b) or 'girl' (g). For three births, the sample space consists of all possible sequences of boys and girls (e.g., ggb, bgg, etc.).

Three Common Approaches to Finding Probability

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

• Relative Frequency Approximation: Probability is estimated by conducting or observing a procedure and calculating the proportion of times event A occurs.

• Classical Approach: If all outcomes are equally likely, the probability of event A is the number of ways A can occur divided by the total number of possible outcomes.

• Subjective Probability: Probability is estimated based on knowledge or experience when historical data or equally likely outcomes are not available.

Notation: P(A) denotes the probability of event A occurring.

Simulations

When none of the three approaches above are feasible, simulations can be used. A simulation is a process that mimics the real procedure to estimate probabilities by generating similar results.

Rounding Probabilities

Probabilities should be expressed as exact fractions or decimals, or rounded to three significant digits for clarity. For example, a probability of 0.183 means the event occurs in about 18.3% of cases.

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 approaches the actual probability. This law applies only to large numbers of trials, not individual outcomes.

• Caution: Do not assume outcomes are equally likely without justification.

Examples of Probability Calculation

• Airline Crashes: If 16 out of 39 million flights crash, the probability of a crash is $P = \frac{16}{39,000,000}$.

• Survey Example: If 366 out of 2003 adults report seeing a ghost, $P = \frac{366}{2003} \approx 0.183$.

Complementary Events

The complement of event A, denoted as A', consists of all outcomes where A does not occur. The probability of the complement is:

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

Example: If 1782 out of 2002 adults use the Internet, the probability that a randomly selected adult does not use the Internet is $P = 1 - \frac{1782}{2002} = \frac{220}{2002} \approx 0.110$.

Identifying Significant Results with Probabilities

The rare event rule states that if, under a given assumption, the probability of an observed event is very small (typically 0.05 or less), and the event occurs, the assumption is likely incorrect.

• Significantly High: x or more successes is significant if $P(x \text{ or more}) \leq 0.05$.

• Significantly Low: x or fewer successes is significant if $P(x \text{ or fewer}) \leq 0.05$.

Additional info: The threshold of 0.05 is commonly used but not absolute; context may require a different threshold.

Probability Review

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

• Notation: P(A) for the probability of event A.

Odds

Odds provide another way to express the likelihood of an event, commonly used in gambling and lotteries.

• Actual Odds Against: The ratio of the probability that event A does not occur to the probability that it does occur.

• Actual Odds In Favor: The reciprocal of the odds against.

• Payoff Odds: The ratio of net profit to the amount bet.

Example: Roulette

• If the probability of winning on number 13 is $\frac{1}{38}$, the actual odds against are 37 to 1.

• Casino payoff odds may be 35 to 1, so a $5 bet yields $175 profit (plus the original $5 bet).

• If the casino paid fair odds (37 to 1), the profit would be $185 for a $5 bet.

Summary Table: Probability Approaches

Approach	Description	Formula	When to Use
Relative Frequency	Estimate based on observed outcomes	$P(A) = \frac{\text{Number of times A occurs}}{\text{Total number of trials}}$	When historical data is available
Classical	Assumes equally likely outcomes	$P(A) = \frac{s}{n}$	When all outcomes are equally likely
Subjective	Based on knowledge or intuition	N/A	When data is unavailable or outcomes are not equally likely