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 impossibility and 1 indicates certainty. Understanding probability allows statisticians to interpret outcomes and make informed decisions based on data.

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

• Simple Event: An outcome that cannot be broken down further.

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

Example: Simple Events and Sample Spaces

• One birth: "girl" or "boy" are simple events.

• Three births: "ggb" (two girls followed by a boy) is a simple event.

• "2 girls and 1 boy" is not a simple event, as it can occur in multiple ways (ggb, gbg, bgg).

• Sample space for three births: Eight different simple events.

Three Common Approaches to Finding Probability

There are three main methods to determine the probability of an event:

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

• Possible Values: Probabilities range from 0 (impossible) to 1 (certain).

• Relative Frequency Approximation: Conduct or observe a procedure and count the number of times event A occurs. Probability is approximated as:

• Classical Approach: If a procedure has n equally likely simple events and event A can occur in s ways: Note: Only use this approach if outcomes are equally likely.

• Subjective Probability: Estimated using knowledge of relevant circumstances when historical data or equally likely outcomes are unavailable.

Simulations

Simulations are used when none of the three approaches above are feasible. A simulation mimics the procedure to produce similar results, allowing probabilities to be estimated.

Rounding Probabilities

Probabilities should be expressed as exact fractions or decimals, or rounded to three significant digits for clarity.

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 its actual probability.

• Applies to large numbers of trials, not individual outcomes.

• Do not assume outcomes are equally likely without evidence.

Example: Airline Crashes

• Relative frequency approach: 16 crashes out of 39 million flights.

• Classical approach not suitable due to unequal likelihood.

• Subjective probability may be used if historical data is lacking.

Example: Ghosts!

• Survey: 366 "yes", 1637 "no", 2003 total responses.

• Probability (relative frequency):

• Interpretation: 0.183 probability that a randomly selected adult reports seeing a ghost.

Complementary Events

The complement of event A, denoted as , consists of all outcomes where A does not occur.

Example: Internet Users

• Survey: 1782 use Internet, 220 do not, out of 2002 adults.

• Probability of not using Internet:

Identifying Significant Results with Probabilities

Statisticians use probabilities to determine whether observed results are significantly high or low, applying the rare event rule for inferential statistics.

• Significantly High: x successes among n trials is significantly high if

• Significantly Low: x successes among n trials is significantly low if

• Note: 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: The ratio "a to b" where a and b are integers, reduced to lowest terms.

• Actual Odds In Favor: If odds against are a:b, odds in favor are b:a.

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

Example: Actual Odds Versus Payoff Odds

• Roulette: Probability of winning by betting on 13 is .

• Casino payoff odds: 35:1. For a $5 bet, net profit is $175; total collected is $180.

• If casino were fair (not for profit), payoff odds would be 37:1, net profit for $5 bet would be $185.