PowerPoint Presentation

Math 361

Probability & Statistics

1Terms & DefinitionsFor any random phenomenon, each attempt, or trial, generates an outcome

So a trial is a single attempt or realization of a random phenomenon

Something happens on each trail, and we call whatever happens the outcome

These outcomes are individual possibilities, such as the number we see on top when we roll a dieTerms & DefinitionsThe outcome of a trial is the value measured, observed, or reported for an individual instance of that trial

Outcomes are considered to be eitherDiscrete if they have distinct values such as heads or tailsContinuous if they take on numeric values in some range of possible valuesTerms & DefinitionsOften, instead of individual possibilities, we want to talk about combinations of outcomes such as The number on the die is less than 4 (that is, it is 1, 2, or 3)

We call such a combination or collection of outcomes an event

Usually, we identify events so that we can attach probabilities to them and denote events with bold capital letters such as A, B, or CTerms & DefinitionsIn order to think about what happens with a series of trials, it really simplifies things if the individual trials are independent

This means that the outcome of one trial does not influence or change the outcome of another

In other words, two events are independent if knowing whether one event occurs does not alter the probability that the other event occurs

In order for us to make statements about the long-run behavior of random phenomena, the trials have to be independentFive Basic Rules of ProbabilityRule 1: Probability RangeA probability is a number between 0 and 1

For any event A, 0 P(A) 1

If the probability is 0, the event never occurs impossible event

If the probability is 1, the event always occurs certain event

Even if you think an event is very unlikely, its probability cant be negative, and even if you are sure it will happen, its probability cant exceed 1Rule 1: Probability ScaleIn the traffic light example we cant record more red lights than the number of times we have observed (or fewer than none)

0 probability means for observed 0% of the times (physically impossible not just unlikely)

1 probability means observed 100% of the times (physically certain not just very likely)Rule 2: Something Has to Happen RuleGenerally any random phenomenon would have more than one outcomes

So we need to distribute the probabilities among all the outcomes a trial can have

How can we do this so that it makes sense?Rule 2 : Something Has to Happen RuleFor example, consider the traffic light example for three consecutive days you may run into a red light all three days or any two days or any one day out of three or none at all

When we assign probabilities to the above outcomes, the first thing to be sure of is that we distribute all of the available probability to all the outcomes

Something always occurs, so the probability of something happening (that is, total probability of all outcomes) is 1Rule 2 : Something Has to Happen RuleWe put all the possible outcomes into a big event called the sample space

This gives us the Something Has to Happen Rule

The probability of the set of all possible outcomes of a trial must be 1: P(S) = 1 (S represents the set of all possible outcomes)

If that is not the case, then there is something wrong or missingRule 3 : Complement RuleSuppose the probability that you get to class on time is 0.8. Whats the probability that you dont get to class on time?

0.2!

The set of outcomes that are not in the event A is called the complement of A, and is denoted by either Ac or A

This leads to the Complement Rule

The probability of an event occurring is 1 minus the probability that it doesnt occur P(A) = 1 P(Ac)Rule 4 : Addition RuleSuppose the possibility that a randomly selected student is a sophomore (A) is 0.20, and the probability that he/she is a junior (B) is 0.30. What is the probability that the student is either a sophomore or a junior, written P(A or B)?

Here we apply the addition rule which says that you can add the probabilities of events that are disjoint (mutually exclusive)

To see whether two events are disjoint, we take them apart into their component outcomes and check whether they have any outcomes in commonRule 4 : Addition RuleDisjoint or mutually exclusive events have no outcomes in common

The Addition Rule states:

For two disjoint events A and B, the probability that one or the other occurs is the sum of the probabilities of the two eventsP(A or B) = P(A) + P(B) provided that A and B are disjointRule 5: Multiplication RuleLets suppose the traffic light spends 35% of its time red and the other 65% green or amber

Whats the chance of finding it red two days in a row?

For independent events, the answer is very simple (and the color of the light today is independent of the color yesterday)

The multiplication rule says that for independent events, to find the probability that both events occur, we just multiply the probabilities togetherRule 5: Multiplication RuleFormally:For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events

P(A and B) = P(A) x P(B)

Provided that A and B are independentRule 5 : Multiplication RuleThis rule can be extended to more than two independent events

Whats the chance of finding the light red every day this week (5 days)?

We can multiply the probability of it happening each day, which is: 0.35 x 0.35 x 0.35 x 0.35 x 0.35 = 0.00525

Or about 5 times in a thousand assuming that all five events are independentDisjoint vs IndependentDisjoint events are mutually exclusive and cannot occur at the same time the occurrence of one excludes the other

Independent events are those that do not affect the probability of each other with their outcome

Disjoint (mutually exclusive) events cannot be independentAll 5 Rules - SummaryProbability is a number between 0 and 1: 0 P(A) 1Something Has to Happen Rule: The probability of the set of all possible outcomes of a trial must be 1 P(S) = 1Complement Rule: The probability of an event occurring is 1 minus the probability that it doesnt occur P(A) = 1 P(Ac)Addition Rule: P(A or B) = P(A) + P(B) provided that A and B are disjointMultiplication Rule: P(A and B) = P(A) x P(B) provided that A and B are independent

What can go wrong?Beware of probabilities that do not add up to oneTo be a legitimate probability distribution, the sum of the probabilities for all possible outcomes must total 1

If the sum is less 1, you may need to add another category and assign the remaining probability to that outcome

If the sum is more than 1, check that the outcomes are disjoint

If they are not then you cannot assign probabilities by just counting relative frequenciesDont Add Probabilities of events if They are Not DisjointEvents must be disjoint to use the Addition Rule

The probability of being under 80 or a male is not the probability of being under 80 plus the probability of being a male

That sum may be more than 1Dont Multiply Probabilities of Events if Theyre Not IndependentThe probability of selecting a student at who is over 610 tall and on the basketball team is not the probability of the student is 610 tall times the probability hes on the basketball team

Knowing that the student is over 610 changes the probability of his being on the basketball team

You cant multiply these probabilities

The multiplication of probabilities of events that are not independent is one of the most common errors people make with probabilitiesDont Confuse Disjoint and IndependentDisjoint events cant be independent

E.g.: If A = {you get an A in this class} and B = {you get a B in this class}, A and B are disjoint

Are they independent?

If you find out that A is true, does that change the probability of B?

You bet if does, so they cant be independentThe End