Probability Questions
explain the difference between the union and the intersection of two or more events
11. Explain the difference between the union and the intersection of two or more events. The union is the event that any of them occur, the intersection is the event that they all occur.
what is an event
An event is any outcome or a set of outcomes of a random phenomenon. That is, an event is subset of the sample space
explain why the probability of any event is a number between 0 and 1
An event with probability 0 never occurs. An event with probability 1 occurs on every trial. The probability of any event must be between these two possible outcomes.
explain the differences between the rules in #2 and #3
If A and B are disjoint, the event {A and B} that both occur has not outcomes in it. This empty event is the complement of the sample space S and must have probability 0. So the general addition rule #3 includes #2, the addition rule for disjoint events
what is the probability of two disjoint event
If A and B are disjoint. Then P(A or B) = P(A) + P(B). This is the addition rule for disjoint events
how is the general multiplication rule different than the multiplication rule for the independent events
If A and B are independent, then P(B|A) = P(B) so the two rules are the same. If A and B are not independent, the rule must be adjusted for this condition
state the addition rule for disjoint events
If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then P(one or more of A, B, C) = P(A) + P(B) + P(C).
can disjoint events be independent
No, disjoint events cannot be independent. If A and B are disjoint, then the fact that A occurs tells us that B cannot occur
what is meant by the complement of an event
The complement of any event A is the event that A does not occur, written as . The complement rule states that P(A^c) = 1-p(A)
what is meant by conditional probability
The conditional probability P(B|A) is the probability that event B occurs given that event A has already occurred
what is meant by the intersection of two or more events?
The intersection of any collection of events is the event that all of the events occur.
what is meant by the union of two or more events?
The union of any collection of events is the event that at least one of the collection occurs.
what is meant by an independent trial
Trials are independent if the outcome of one trial does not influence the outcome of any other.
when are two events considered disjoint?
Two events A and B are disjoint (also called mutually exclusive) if they have no outcomes in common and so can never occur simultaneously.
what is the multiplication rule for independent events
Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, P(A and B) = P(A)P(B).
state the formula used to determine if two events are independent
Two events A and B that both have positive probability are independent if P(B|A) = P(B).
In statistics, what Is meant by the term random?
We call phenomenon random if individual outcomes are uncertain but there in nonetheless a regular distribution of outcomes in a large number of repetitions. "Random" in statistics is not a synonymfor "haphazard" but a description of a kind of order that emerges only in the long run
state the formula for finding conditional probability
When P(A) > 0, the conditional probability of B given A is: P(B|A) = p(A and B)/P(A)
state the general addition rule for unions of two events
For any two events A and B, P(A or B) = P(A) + P(B) - P(A and B).
if two events A and B are independent, what must be true about A^c and B^c
If A and B are independent then Ac and Bc are also independent and Ac is independent of B.
what is the sum of the probabilities of all possible outcomes
Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly 1.
explain why the probability of getting heads when flipping a coin is 50%
P(A)= (count of outcomes in A/count of outcomes in S)= (count of outcomes in A/K)
what is probability theory
Probability theory is the branch of mathematics that describes random behavior.
what is meant by probability
The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is long-term relative frequency
describe the probability that an event does not occur
The probability that an event does not occur is 1 minus the probability that the event does occur. The probability that an event occurs and the probability that it does not occur always add to 100% or 1.
state the general multiplication rule
The probability that both of two events A and B happen together can be found by P(A and B) = P(A)P(B|A)
what is a sample space
The sample space S of a random phenomena is the set of all possible outcomes.
what is meant by joint probability
The simultaneous occurrence of two events is called a joint event. The probability of a joint event is called a joint probability