stats chapter 5
conditional probability.
Suppose we know that event A has happened. Then the probability that event B happens given that event A has happened is denoted by P(B | A).
General Addition Rule for Two Events
If A and B are any two events resulting from some chance process, then P(A or B) = P(A) + P(B) - P(A and B)
Addition rule for mutually exclusive events:
If A and B are mutually exclusive, P(A or B) = P(A) + P(B).
The general multiplication rule states that the probability of events A and B occurring together is
P(A ∩ B)=P(A) • P(B|A)
Complement rule:
P(AC) = 1 - P(A)
The conditional probability formula states
P(B|A) = P(A ∩ B) / P(A)
general multiplication rule
The idea of multiplying along the branches in a tree diagram leads to a general method for finding the probability P(A ∩ B) that two events happen together. P(A ∩ B) = P(A) • P(B | A) where P(B | A) is the conditional probability that event B occurs given that event A has already occurred.
All probability models must obey the following rules
The probability of any event is a number between 0 and 1. All possible outcomes together must have probabilities whose sum is 1. If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula : P(A)= Number possible outcomes corresponding to event A / Number possible outcomes in the sample space The probability that an event does not occur is 1 minus the probability that the event does occur. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities
Multiplication rule for independent events
When events A and B are independent, we can simplify the general multiplication rule since P(B| A) = P(B).
A probability model is
a description of some chance process that consists of two parts: a sample space S and a probability for each outcome.
When finding probabilities involving two events,
a two-way table can display the sample space in a way that makes probability calculations easier.
The union (A ∪ B) of events A and B consists of
all outcomes in event A, event B, or both.
An event is
any collection of outcomes from some chance process. That is, an event is a subset of the sample space. Events are usually designated by capital letters, like A, B, C, and so on.
A two-way table or a Venn diagram can be used to
display the sample space for a chance process.
Events A and B are independent if
if the chance that event B occurs is not affected by whether event A occurs. If two events are mutually exclusive (disjoint), they cannot be independent.
The intersection (A ∩ B) of events A and B consists of
outcomes in both A and B.
The law of large numbers
says that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value.
Two events A and B are independent
the occurrence of one event has no effect on the chance that the other event will happen. In other words, events A and B are independent if P(A | B) = P(A) and P(B | A) = P(B).
conditional probability. P(B|A) represents
the probability that event B occurs given that event A has occurred.
The intersection of events A and B (A ∩ B) is
the set of all outcomes in both events A and B.
The union of events A and B (A ∪ B) is
the set of all outcomes in either event A or B.
The sample space S of a chance process is
the set of all possible outcomes.
Two events are mutually exclusive (disjoint) if
they have no outcomes in common and so can never occur together.
