Bist P6104_Probability

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

• Pr (A and B occur) = Pr (A) X Pr (B)
• under independence.

If event A is independent of B, then A^c is also independent of B.

"At least one ___ " is the complement of "none" (consider K events = A₁, A₂, .... A_k)

The event that {at least one of the A₁, A₂, .... A_k occurs} is the complement of {none A₁, A₂, .... A_k occurs}

Pr (at least 1) = 1-Pr(at least 1)^c

Pr(atleast 1)^c = p(none)

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.

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

A union B = A or B =

A U B

A intersect B = A and B =

A (upside down U) B

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

Suppose A and B are dependent events then, the occurrence of A provides information about the likelihood of the occurrence of B.

The conditional probability of an event A given that B occurs is denoted by

Pr (A|B) and is defined as

Pr (A|B) = Pr (A and B)/Pr (B)

Pr (A and B) = Pr (A|B) Pr (B)

Pr (A|B) = Pr (A)

if A and B are independent.

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.

The total probability rule

Pr (A) = Pr (A|B) X Pr (B) + Pr (A|B^c) X Pr (B^c)

Pr (A) = Pr( A and B) + Pr (A and B^c)

Pr (B|A) = Pr (A|B) X Pr (B)/Pr(A|B) X Pr (B) + Pr(A|B^c) x Pr (B^c)

Pr (B|A) = Pr (A and B)/Pr (A and B) OR Pr (A and B^c)

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

• 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

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

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

Pr (A U B) = Pr (A) + Pr (B) X [1 - Pr (A)]

Pr (A U B ) = Pr (A) + Pr (B) X Pr (A^c)

The probability that a test is positive for a symptom given that the person has the symptom.

Sensitivity =Pr (Test=Yes|person has disease)

The probability that a test scores negative for a symptom given that the person does not have the symptom.

Specificity = Pr (Test = No|person has no disease)

The probability that a person has the disease given that the test scored positive for the disease.

Predictive value positive = Pr (person has disease|test is positive)

The probability that a person is not diseased given that the test scored negative for the disease.

Predictive value negative = Pr (person has no disease|test = negative)