PROBABILITY ESSENTIALS

• Concept of probability is quite intuitive; however, the rules of probability are not always intuitive or easy to master.

• Mathematically, a probability is a number between 0 and 1 that measures the likelihood that some event will occur.

– An event with probability zero cannot occur.

– An event with probability 1 is certain to occur.

– An event with probability greater than 0 and less than 1 involves uncertainty, but the closer its probability is to 1 the more likely it is to occur.

Probability•Probability plays a key role in statistics

•Probability theory has two basic building blocks.

–The more fundamental is the generating mechanism, called the random experiment, that gives rise to uncertain outcomes

For example, selecting one part for inspection from an incoming shipment is a random experiment having two elements of uncertainty: which particular item gets picked and the quality of that part.

Probability

• The second structural element of probability theory consists of the random experiment’s outcomes, referred to as events

For example: during a quality-control inspection, the usual events of interest for tested items are “goods” and “defective”

Probability• Elementary Events

An example: If we are interested only in the particular upside showing face from a coin toss, then we have

Coin toss sample space = {head, tail} (showing face)

A preliminary step in a probability evaluation is to catalog the events that might arise from the random experiment.

Such a listing is made up of elementary events, which are the most detailed events of interest.

The immediate concern when finding a probability is properly identifying the event. Any subset of a sample space is called event. By subset we mean any part of a set.

Sample Space

• In statistics, a set of possible outcomes of an experiment is called sample space. Sample spaces are usually denoted by the letter S.

• A sample space might be portrayed as a list, as above, or in some other convenient form.

• Sometimes it is conceptually helpful to use a picture

Probability

The probability of an event (happening or outcome) is the proportion of times the event would occur in a long run of repeated experiments.

For example, if we say that the probability is 0.78 that a jet from New York to Boston will arrive on time, we mean that such flights arrive on time 78% of the time.

Example

• If records show that 294 of 300 ceramic insulators tested were able to withstand a certain thermal shock, what is the probability that any one such insulator will be able to withstand the thermal shock?

Solution: Among the insulators tested, 294/3000 = 0.98 were able to withstand the thermal shock

Basic Definitions of Probability• Long-run Frequency: If a perfectly balanced

coin is tossed many times without bias toward either side, we should obtain a head in about half the tosses.

• Thus, the long-run frequency of “head” is 0.50. We may then assume that 0.50 is the probability of “head” expressed symbolically as

Pr[head] = 0.50

Objective Probability

• The previous value is an objective probability, and there should be no agreement about how to find its value.

• Objective probabilities can often be found, as above, through deductive reasoning alone.

• We don’t actually have to toss the coin to reach the answer since we have no reason to believe that the head side will show any more or less than half the time.

Probability for an event

If elements are equally likely, Probability for an event can be deduced directly,

Pr[event] = size of the event/ size of sample space

Certain Event

• An outcome bound to occur is a certain event

• It has a probability of 1.

Impossible Event

• An outcome having zero probably is called an impossible event, since it cannot occur.