Math Chapter 4

The probability of one event occurring, given that another event has occurred is called a

conditional probability.

If we can replace the first item drawn before sampling the second item, is known as

sampling with replacement it is possible to the same item more than once. When sampling with replacement, the sampled items are independent.

If we leave the first item out when sampling the second one is known as

sampling without replacement. When sampling without replacement, if the sample size is less than 5% of the population, the sampled items may be treated as independent. When sampling without replacement, if the sample size is more than 5% of the population, the sample items cannot be treated as independent

with replacement

a times b

The probability of an event is always between If event A cannot occur, then If event A is certain to occur,

-0 and 1 -P(A)=0 3. -the P(A)=1

Probability rule 1:Any probability is a number between 0 and 1 0<P(A)<1

-A probability can be interpreted as the proportion of times that a certain event can be expected to occur. -If the probability of an event is more than 1, then it will occur more than 100% of the time (Impossible!).

Probability 4: The probability that an event does not occur is 1 minus the probability that the event does occur. P(A does not occur)= 1- P(A)

-As a jury member, you assess the probability that the defendant is guilty to be 0.80. Thus you must also believe the probability the defendant is not guilty is 0.20 in order to be coherent (consistent with yourself). -If the probability that a flight will be on time is .70, then the probability it will be late is .30.

Probability 2: All possible outcomes together must have probability 1. P(A)=1

-Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly one. -If the sum of all of the probabilities is less than one or greater than one, then the resulting probability model will be incoherent.

Note : P(Atleast one) =

1- P( None)

Venn diagrams

A picture that shows the sample space S as a rectangular area and events as areas within S is called a Venn diagram.

Event

Any collection of results or outcomes of a procedure

Independent

Conditions probability of A given B would be same as probability of A in general

Middle Left Right Outer

D and T=0.44 D and not T=0.33 T and not D=0.08 Neither D nor T=0.15

Multiplication rule

For ANY two events, the probability that they both happen is found by multiplying the probability of one of the events by the conditional probability of the remaining event given that the other occurs:

General addition rule

Out comes in middle are are double -counted by P(A) + P(B) P(A or B) = P(A) + P(B) P(A and B)

Simple event

Outcome or an event that cannot be further broken down into simpler components.

The Multiplication rule extends to find the probability that all of the several events occur

P( A and B and C)= P(A) P(B | A) P(C | both A and B)

Multiplication Rule for independent events:

P(A and B) = P(A) P(B)

General Multiplication Rule

P(A and B) = P(A) P(B|A)

Multiplication rule 2

P(A and B) = P(A) P(B|A) or P(A and B) = P(B) P(A|B)

Mutually exclusive: Not occurring at same time

P(A and B)=0 P(A or B)= P(A)+P(B)

Disjoint events

P(A or B) = P(A) + P(B)

Not mutually exclusive: occurring at same time

P(A or B)= P(A)+P(B)-P(A and B)

If a sample space has n equally likely outcomes, and an event A has k outcomes,then

P(A)= # of outcomes in A/ Number of outcomes in sample space= k/n

The conditional probability of B given A is denoted by

P(B|A) the proportion of all occurrences of A for which B also occurs

Two events A and B that both have positive probability are independent if

P(B|A) = P(B)

When P(A) > 0, the conditional probability of B given A is

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

P is

Probability

P(A)

Probability of event occurring

Sample space

Procedure consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further

A, B, C

Specific events

Law of large numbers

The Law of large numbers says that as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability.

Probability

The probability of an event is the proportion of times the event occurs in the long run, as a probability experiment is repeated over and over again.

Two events that are not disjoint, and the event {A and B} consisting of the outcomes they have in common

Two connected circles

Independent events

Two events are independent if the occurrence of one does not affect the probability that the other event occurs. If two events are not independent, we say they are dependent.

Mutually exclusive (dis)

Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint. If two events are disjoint, then the probability of them both occurring at the same time is 0.

Two disjoint vents

Two separate circles

Disjoint

Two separate events would never occur together P(A or B) is used if they are disjoint

Tree diagram

Useful for solving probability problems that involve several stages • Often combine several of the basic probability rules to solve a more complex problem -probability of reaching the end of any complete "branch" is the product of the probabilities on the segments of the branch (multiplication rule) -probability of an event is found by adding the probabilities of all branches that are part of the event (addition rule)

A probability model for a probability experiment consists of

a sample space, along with a probability for each event

without replacement

b given a times a

If A is any event , the complement of A is

the event that A does not occur. ( Denoted as A' ) P( A') = 1- P(A)

Probability 3: If two events have no outcomes in common, P(A or B)= P(A) + P(B)

they are said to be disjoint. The probability that one or the other of two disjoint events occurs is the sum of their individual probabilities.