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What is the definition of independent events?
If the occurrence of an event A does not affect or depend on the occurrence of B, then A and B are independent events.

What is the matematical definition of independent events?
 Pr (A and B occur) = Pr (A) X Pr (B)
 under independence.

What is the first principle under independence?
If event A is independent of B, then A^{c }is also independent of B.

What is the second principle under independence?
"At least one ___ " is the complement of "none" (consider K events = A_{1}, A_{2}, .... A_{k})
The event that {at least one of the A_{1}, A_{2}, .... A_{k} occurs} is the complement of {none A_{1}, A_{2}, .... A_{k} occurs}
Pr (at least 1) = 1Pr(at least 1)^{c
}Pr(atleast 1)^{c} = p(none)

What is the definition of the addition law of probability?
The addition law of probability states that the probability of events is the sum of the events minus the multiplication of the events.
The addition law of probability is the additive probability minus the multiplicative probability.

What is the matematical definition of the addition law of probability?
If A and B are any events, then
Pr (A or B occurs) = Pr (A) + Pr (B)  Pr (A and B)
Note: Pr (A and B) = Pr (A) X Pr (B) (under independence)
Note: Pr (A and B) = 0 if mutually exculsive

What is the matematical shortcut for events A or B?
A union B = A or B =
A U B

What is the matematical shortcut for both events A and B occur?
A intersect B = A and B =
A (upside down U) B

Can two events be mutually exclusive AND independent?
No. Two events cannot be mutually exclusive and independent since mutally exclusive events do not occur simultaneously and independent events can occur together.

What is the definition of conditional probability?
Suppose A and B are dependent events then, the occurrence of A provides information about the likelihood of the occurrence of B.

What is the mathematical definition of conditional probability?
The conditional probability of an event A given that B occurs is denoted by
Pr (AB) and is defined as
Pr (AB) = Pr (A and B)/Pr (B)

What is a consequence of the definition of conditional probability?
Pr (A and B) = Pr (AB) Pr (B)

What is the conditional probability of A given B if A and B are independent?
Pr (AB) = Pr (A)
if A and B are independent.

What is the definition of the totalprobability rule?
The probability of Event A is the Probability of event A given B times the probability of B plus the probability of event A given the complement of B times the complement of B.
The probability of event A is the probability of event A and B OR event A and B^{c}.

What is the mathematical definition of the total probability rule?
The total probability rule
Pr (A) = Pr (AB) X Pr (B) + Pr (AB^{c}) X Pr (B^{c})
Pr (A) = Pr( A and B) + Pr (A and B^{c})

What is the mathematical definition of Bayes' rule?
Pr (BA) = Pr (AB) X Pr (B)/Pr(AB) X Pr (B) + Pr(AB^{c}) x Pr (B^{c})
Pr (BA) = Pr (A and B)/Pr (A and B) OR Pr (A and B^{c})

What is a random variable?
A random variable is a function that assigns a number to each element in the sample space.
For example, if the sample space is: sss, sfs, ssf, fss, sff, fsf, ffs, fff}
define a r.v. if x _{i} = {1=if the i ^{th }patient has a response, 0 if else}
 then
 X1 (ssf) =1, X1 (sfs) =1
 X2 (ssf) =1, X2 (sfs) =0

How can we compute the probability of any event?
 The probabililty of any event defined in terms of a discrete r.v. say X, can be computed if we know the distribution of X which can be expressed via its probability mass function (pmf) defined as
 Pr (X=r) for all possible values of r that X can take on

How are the probability of events estimated?
The probability of events are estimated from the empirical probabilities obtained from large samples.

What is an important issue in statistical inference regarding empirical probabilities?
An important issue in statistical inference is to compare empirical probabilities with theoretical probabilities  asses the goodness of fit of probability models.

What is the mathematical definition of the union of two independent events?
Pr (A U B) = Pr (A) + Pr (B) X [1  Pr (A)]
Pr (A U B ) = Pr (A) + Pr (B) X Pr (A^{c})

What is the sensitivity of a test?
The probability that a test is positive for a symptom given that the person has the symptom.
Sensitivity =Pr (Test=Yesperson has disease)

What is the specificity of a test?
The probability that a test scores negative for a symptom given that the person does not have the symptom.
Specificity = Pr (Test = Noperson has no disease)

What is the definition of predictive value positive?
The probability that a person has the disease given that the test scored positive for the disease.
Predictive value positive = Pr (person has diseasetest is positive)

What is the definition of predictive value negative?
The probability that a person is not diseased given that the test scored negative for the disease.
Predictive value negative = Pr (person has no diseasetest = negative)

