Chp 14/15 stats
independent
. Informally, this means that the outcome of one trial doesn't affect the outcomes of the others.independence means that the outcome of one event doesn't influence the outcome of the other.
Event
A collection of outcomes. Usually, we identify events so that we can attach probabilities to them. We denote events with bold capital letters such as A, B, or C.
ndependence (used formally)
Events A and B are independent when P(BlA)=P(B)
independent formal
Events A and B are independent whenever P(BlA)=P(B)Events Aand B are independent if (and only if) the probability of A is the same when we are given that B has occurred.
general multiplication rule
For any two events, A and B, the probability of A and B is P(AnB)=P(A)XP(BlA)
Conditional probability
P(BlA)=P(AnB)/P(A)
general addition rule
We add the probabilities of two events and then subtract out the probability of their intersection. This approach gives us the General Addition Rule, which does not require disjoint events
Independence Assumption
We often require events to be independent. (So you should think about whether this assump- tion is reasonable.
Sample space
We sometimes talk about the collection of all possible outcomes and call that event the sample space.
Theoretical probability
When the probability comes from a model (such as equally likely outcomes), it is called a the-oretical probability.
Empirical probability
When the probability comes from the long-run relative frequency of the event's occurrence, it is an empirical probability
Personal probability
When the probability is subjective and represents your personal degree of belief, it is called a personal probability
general multiplication rule
for compound events that does not require the events to be independent. Better than that, it even makes sense. The probability that two events, A and B, both occur is the probability that event A occurs multiplied by the probability that event B also occurs—that is, by the probability that event B occurs given that event Aoccurs. Of course, there's nothing special about which set we call A and which one we call B. We should be able to state this the other way around.
Empirical probability
is based on repeatedly observing the event's outcome, this definition of probabilty is often called empirical probability.
Trial
single attempt or realization of a random phenomenon.
Tree diagram
A display of conditional events or probabilities that is helpful in thinking through conditioning.
Random phenomenon
A phenomenon is random if we know what outcomes could happen, but not which particular values will happen.
Addition rule
Addition Rule, which says that you can add the probabilities of events that are disjoint. To see whether two events are disjoint, we take them apart into their component outcomes and check whether they have any outcomes in common.
Legitimate probability assignment
An assignment of probabilities to outcomes is legitimate if each probability is between 0 and 1 (inclusive)/the sum of the probabilities is 1.
Don't reverse conditioning naively.
As we have seen, the probability of A given B may not, and, in general does not, resemble the probability of B given A. The true probability may be counterintuitive.
Outcome
At each trial, we note the value of the random phe- nomenon, and call that the trial's outcome.
Legitimate
Because sample space outcomes are disjoint, we have an easy way to check whether the probabilities we've assigned to the possible outcomes are legitimate. The Probability Assignment Rule tells us that the sum of the probabilities of all possible outcomes must be exactly 1. No more, no less
Probability
Because the LLN guarantees that relative frequencies settle down in the long run, we can now officially give a name to the value that they approach. We call it the probability of the event.
Complement rule
Complement Rule:The probability of an event occurringThe set of outcomes that are not in the event A is called the complement of A, and is denoted AC. This is 1 minus the probability that it doesn't occur.
Don't confuse "disjoint" with "independent."
Disjoint events cannot happen at the same time. When one happens, you know the other did not, so P(BlA)=0. Independent events must be able to happen at the same time. When one happens, you know ithas no effect on the other, so P(BlA)=P(B)
Don't use a simple probability rule where a general rule is appropriate.
Don't assume independence without reason to believe it. Don't assume that outcomes are disjoint without checking that they are. Remember that the general rules always apply, even when outcomes are in fact independent or disjoint.
General Addition Rule
For any two events, A and B, the probability of A or B is P(AuB)=P(A)+P(B)-P(AnB)
Addition Rule
If A and B are disjoint events, then the probability of A or B is P(AuB)=P(A)+P(B)
Multiplication Rule
If A and B are independent events, then the probability of A and B is P(AnB)=P(A)*P(B)
A probability is a number between 0 and 1
If the probability is 0, the event can't occur, and like- wise if it has probability 1, it always occurs. Even if you think an event is very unlikely, its probability can't be negative, and even if you're sure it will happen, its probability can't be greater than 1.
Some
In Informal English, you may see "some" used to mean at least one." "What's the probability that some of the eggs in that carton are broken?" means at least one.
Trial
In general, each occasion upon which we observe a random phenomenon is called a trial.
Indi assumption
Independence Assumption, but assuming independence doesn't make it true. Always Think about whether that assumption is reasonable before using the Multiplication Rule.
Theoretical probability
Math models.to find probabilities for events that are made up of several equally likely outcomes. We just count all the outcomes that the event contains. The prob- ability of the event is the number of outcomes in the event divided by the total number of possible outcomes
Multiplication rule
Multiplication Rule says that for independent events, to find the probability that both events occur, we just multiply the probabilities together. Formally,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.can be extended than for more than two indi events
Don't find probabilities for samples drawn without replacement as if they had been drawn with replacement.
Remember to adjust the denominator of your probabilities. This warning applies only when we draw from small populations or draw a large fraction of a finite population. When the population is very large relative to the sample size, the adjustments make very little difference, and we ignore them.
Personal probability
Subjective.personal probabilities may be based on experience, they're not based either on long-run relative frequencies or on equally likely events. So they don't display the kind of consistency that we'll need probabilities to have. For that reason, we'll stick to formally defined probabilities
Lln
The LLN says that as the number of independent trials increases, the long-run relative frequency of repeated events gets closer and closer to a single value.
Law of Large Numbers
The Law of Large Numbers states that the long-run relative frequency of repeated independ- ent events gets closer and closer to the true relative frequency as the number of trials increases.
Sample Space
The collection of all possible outcome values. The sample space has a probability of 1.
tree diagram
The kind of picture that helps us think through this kind of reasoning iscalled a tree diagram,because it shows sequences of events, like those we had in room draw, as paths that look like branches of a tree.
Outcome
The outcome of a trial is the value measured, observed, or reported for an individual instance of that trial.
Probability
The probability of an event is a number between 0 and 1 that reports the likelihood of that event's occurrence. We write P(A) for the probability of the event A.
Complement Rule
The probability of an event occurring is 1 minus the probability that it doesn't occur.
The Probability Assignment Rule
The probability of the entire sample space must be 1. P(S) =1.
Probability assignment rule
The set of all possible outcomes of a trial must have probability 1. If a random phenomenon has only one possible outcome, it's not very inter- esting (or very random). So we need to distribute the probabilities among all the outcomes a trial can have.
Beware of probabilities that don't add up to 1.
To be a legitimate probability assignment, the sum of the probabilities for all possible outcomes must total 1. If the sum is less than 1, you may need to add another category ("other") and assign the remaining probability to that outcome. If the sum is more than 1, check that the outcomes are disjoint. If they're not, then you can't assign probabilities by just counting relative frequencies.
Disjoint (Mutually exclusive)
Two events are disjoint if they share no outcomes in common. If A and B are disjoint, then knowing that A occurs tells us that B cannot occur. Disjoint events are also called "mutually exclusive."
Independence (informally)
Two events are independent if learning that one event occurs does not change the probability that the other event occurs.
Event
When we combine outcomes, the resulting combination is an event. Each individual outcome is also an event.
Axioms
axioms—state- ments that we assume to be true of probability
Disjoint events
or mutually exclusive) events have no outcomes in common. The AdditionRule states,For two disjoint events A and B, the probability that one or the otheroccurs is the sum of the probabilities of the two events.
conditional probability
when we want the probability of an event from a conditional distribution, we write the probability of B given A. A probability that takes into account a given condition/to find the probability of the event B given the event A,we restrict our attention to the outcomes in A. We then find in what fraction of those outcomes B also occurred.