Chapter 4
Addition Rule for mutually exclusive events
-If A and B are mutually exclusive events, then P(A or B)=P(A) + P(B) - prove the P(A and B)=0 in order to prove that the events are mutually exclusive
Probability Rules
-The probability of an event is always between 0 and 1. -If A cannot occur, then P(A)=0-> Impossible event -If A is certain to occur, then P(A)=1 -> certain event -The sum of the probabilities of all the outcomes in a sample space is 1.
The conditional probability of an event B given/ in relationship to an event A is the probability that event B occurs, under the assumption that A occurs, in other words AFTER event A has already occurred.
-denoted P(B|A), read as "the probability of B given A" -The outcome of A (event right on the vertical line) is already known. -The conditional probability is computes as P(B|A) = P(A and B)/P(A)
General Addition Rule
Used to compute probabilities of the form P(A or B) -The event A or B occurs whenever A occurs, B occurs, Or A and B both occur -P(A or B)= P(A)+ P(B)-P(A and B)
Event
a specific collection of outcomes from a sample spaced. ex: rolling an odd number -> outcomes {1,3,5}
When two events are mutually exclusive, then P(A and B)=0
The leads to the simplification of the general addition rule -> Addition Rule for mutually exclusive events
The Law of Large Numbers
Another way to state our definition of probability: as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability.
The Empirical Method
Allows us to approximate the probability of an event by repeating a probability experiment many times and computing the proportion of times the event occurs.
The complement of an event A is denoted Ac, is the event that A does not occur
Exactly (=) a number-> Not equal the number More than (>) a number-> The number of fewer (less than or equal to) At least (greater than or equal to) a number -> fewer than (<) the number
A compound event
Is an event that is formed by combining two or more events Ex: A or B
If A and B are independent, then P(B|A)=P(B) also, P(A|B)=P(A).
Leads to the simplification of the genreal multiplication rule-> Multiplication Rule for independent events
Multiplication Rule for Independent events
P(A and B)=P(A)*P(B) To determine whether two events are independent prove that P(A and B)=P(A)*P(B)
Computing Probabilities with equally likley outcomes:
P(A)= Number of outcomes in A/ Number of outcomes in the sample space = k/n
The Rule of complements
P(A^c)=1-P(A)
A probability model
Specifies a probability for every event, and must follow the probability rules
Outcome
The result of a single trial of a probability experiment.
When sampling from a population, sampled individuals are independent if the sampling is done with replacement, or if the sample size is less than 5% of the population
True.
Independent
Two events A and B are independent if the fact that A occurs does NOT affect the probability of B occurring
Mutually exclusive
Two events are mutually exclusive if it is impossible for both events to occur -Ask yourself if the occurrence of one will prevent the other from occurring.
Probability experiment
an experiment that can result in any one of a number of outcomes ex: Tossing a fair coin or rolling a die are examples of a probability experiment.
A _______ probability is a probability that is computed with the knowledge of additional information
conditional
General multiplication rule is used to compute probabilities of the for P(A and B)
for any two events A and B that are NOT independent, P(A and B) =P(A) * P(B|A) or P(A and B)= P(B)*P(A|B)
An unusual event
one whose probability is small. An event A is unusual if P(A) < 0.05
Sample space
the collection of all possible outcomes
The probability of an event
the proportion of times the event occurs in the long run, as a probability experiment is repeated over and over again. If A denotes an event, P(A) denotes the probability of event A
