Chapter 4: Probability – Basic Concepts and Applications

Probability: Basic Concepts

Introduction to Probability

Probability is a fundamental concept in statistics that quantifies the likelihood of events occurring. Probabilities are always expressed as values between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

• Key Point: Students should be able to calculate and interpret probabilities, and understand how odds relate to probabilities, especially in contexts such as lotteries and gambling.

Basic Definitions of Probability

Events, Simple Events, and Sample Space

• Event: Any collection of results or outcomes from a procedure. Denoted by capital letters (e.g., A, B).

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

• Sample Space (S): The set of all possible simple events for a procedure. It contains all outcomes that cannot be broken down further.

Example:

• Sample space for tossing a coin: S = {Heads, Tails}

• Sample space for rolling a die: S = {1, 2, 3, 4, 5, 6}

Complementary Events

• Complement: The complement of event A, denoted by A', consists of all outcomes in which event A does not occur.

Example: Let A be the event of getting at least 3 when rolling a die. Then A = {3, 4, 5, 6}. The complement, A', is {1, 2}.

Approaches to Finding Probability

Three Common Approaches

There are three main approaches to finding the probability of an event, all resulting in values between 0 and 1:

1. Relative Frequency Approximation: Conduct (or observe) a procedure and count the number of times event A occurs. Probability is approximated as: $P(A) = \frac{\text{Number of times A occurred}}{\text{Number of trials}}$

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

3. Subjective Probability: Probability is estimated using knowledge of relevant circumstances. This estimate may be accurate or inaccurate depending on the information available.

Additional info: Simulations can be used when none of these three procedures are applicable, especially in complex or non-repeatable scenarios.

Example: Relative Frequency – Skydiving

• In a recent year, there were about 3,000,000 skydiving jumps and 21 resulted in death.

• Using the relative frequency approach: $P(\text{skydiving death}) = \frac{21}{3,000,000} \approx 0.000007$

• The classical approach cannot be used here because the outcomes (dying, surviving) are not equally likely.

Example: Sample Space and Events with Months

• Sample space S for drawing a slip from a jar with each month: S = {January, February, March, April, May, June, July, August, September, October, November, December}

• Event A: Drawing a month beginning with 'J' = {January, June, July}

• Event B: Drawing a month with exactly four letters = {June, July}

Example: Drawing Playing Cards

• An ordinary deck has 52 cards: 4 suits (spades, hearts, diamonds, clubs), 13 cards per suit.

• Spades and clubs are black; diamonds and hearts are red.

• Probability of drawing a spade: $P(\text{spade}) = \frac{13}{52} = \frac{1}{4}$

• Probability of drawing a club: $P(\text{club}) = \frac{13}{52} = \frac{1}{4}$

Law of Large Numbers

Definition and Implications

• As a procedure is repeated many times, the relative frequency probability of an event tends to approach the actual probability.

• Cautions:

  • The law applies only over a large number of trials, not to individual outcomes.

  • If nothing is known about the likelihood of outcomes, do not assume they are equally likely.

Odds

Types of Odds

• Actual Odds Against: The ratio $\frac{P(A')}{P(A)}$, usually expressed as a:b, where a and b are integers reduced to lowest terms.

• Actual Odds in Favor: The reciprocal of the odds against, or $\frac{P(A)}{P(A')}$; if odds against are a:b, odds in favor are b:a.

• Payoff Odds: The ratio of net profit to the amount bet, expressed as (net profit):(amount bet).

Example: Actual Odds vs. Payoff Odds in Roulette

• Probability of winning by betting on 13: $P(13) = \frac{1}{38}$; $P(\text{not 13}) = \frac{37}{38}$

• Actual odds against 13: $\frac{37}{1}$ or 37:1

• Casino payoff odds: 35:1 (for every $1 bet, $35 profit)

• For a $5 bet, net profit at casino odds: $5 \times 35 = $175; total collected = $180 (including original bet)

• If casino used actual odds (37:1), net profit for $5 bet would be $185

Type of Odds	Formula	Example (Roulette, betting on 13)
Actual Odds Against	$\frac{P(\text{not 13})}{P(13)}$	37:1
Payoff Odds (Casino)	(Net profit):(Amount bet)	35:1
Net Profit for $5 Bet (Casino)	$5 \times 35$	$175
Net Profit for $5 Bet (Actual Odds)	$5 \times 37$	$185

Additional info: The difference between actual odds and payoff odds represents the casino's profit margin.