Chapter 4 - Probability and Probability Distributions Flashcards
As n gets larger… Sample -> ? Relative Frequence -> ?
Sample -> Population Relative Frequency -> Probability
The process by which an observation (or measurement) is obtained
Experiment
The outcome that is observed on a single repetition of the experiment
Simple Event
A Simple Event is denoted by?
E with a subscript
Each simple event will be assigned a __ measuring __ ___ it occurs.
Probability How Often (How Likely)
The set of all simple events of an experiment
Sample Space, S
A collection of one or more simple events (a subset of sample space)
an Event
Two events are __ __ if, when one event occurs, the other cannot, and vice versa.
mutually exclusive
The probability of an event A measures “how often” we think A will occur. We write?
P(A)
If we let n get infinitely large
P(A) = lim(n->∞) f/n
P(A) must be between _ and _
0 and 1
The sum of the probabilities for all simple events in S equals __.
1
The probability of an event A is found by..
adding the probabilities of all the simple events contained in A.
If the simple events in an experiment are equally likely, you can calculate… You can use ___ ___ to find n(a) and N
P(A) = n(a) / N Counting Rules
If an experiment is performed in two stages, with m ways to accomplish the first stage and n ways to accomplish the second stage, then…
There are mn ways to accomplish the experiment
The mn Rule is easily extended to k stages with the number of ways equal to..
n(1)n(2)n(3)…n(k)
The number of ways you can arrange
n distinct objects, taking them r at a time
Permutations
The number of distinct combinations of n distinct objects that can be formed, taking them r at a time
Combinations
A ∪ B
The UNION of two events, A and B
The Union of two events, A and B is the event that..
We write…
Either A or B or BOTH occur when the experiment is performed.
A ∪ B
A ∩ B
the Intersection of two events, A and B
the Intersection of two events, A and B
We write..
Is the event that both A and B occur when the experiment is performed
A ∩ B
If two events A and B are mutually exclusive then P(A ∩ B) = ?
P(A ∩ B) = 0
What consists of all outcomes of the experiment that do not result in event A?
We write?
The Complement of an event A
Ac
The Additive Rule for Unions
For any two events, A and B, the probability of their union P(A∪B) is
P(A∪B) = P(A) +P(B) - P(A∩B)
When two events A and B are mutually exclusive, P(A∩B) = 0
and?
P(A∪B) = P(A) + P(B)
P(A ∩ Ac) =
0
P(A ∪ Ac) =
1
P(Ac) =
1-P(A)
The rule for calculating P(A B) depends on the idea of ___ and ___ events
Independant, Dependant
Two events, A and B, are said to be independent if and only if
the probability that event A occurs does not change, depending on whether or not event B has occurred.
The probability that A occurs, given that event B has occurred is called the ___ ___ of A given B
conditional probability
Conditional Probability of A given B is defined as
P(A | B)
the line seperating A and B means
“given”
Two events A and B are independent if and only if
P(A|B) = P(A)
AND
P(A∩B) = P(A)P(B)
The Multiplicative Rule for Intersections
For any two events, A and B, the probability that both A and B occur is…
P(A∩B) = P(A) P(B|A)
= P(B) P(A|B)
If the events A and B are independent, then the probability that both A and B occur is
P(A∩B) = P(A) P(B)
Law of Total Probabilty
Let S1, S2, S3,…,Sk be mutually exclusive. Then the probability of another event A can be written as..
P(A) = P(A∩S1) + P(A∩S2) +…+P(A∩Sk)
= P(S1)P(A|S1) +P(S2)P(A|S2) +…+P(Sk)(P(A|Sk)
Bayes Rule
Let S1 , S2 , S3 ,…, Sk be mutually exclusive and exhaustive events with prior probabilities P(S1), P(S2),…,P(Sk). If an event A occurs, the posterior probability of Si, given that A occurred is
A quantitative variable x is a __ ___ if the value that it assumes,
corresponding to the outcome of an experiment is a chance or random event.
Random Variable
Random variables can be __ or ___.
discrete or continuous
The probability distribution for a discrete random variable x is a graph, table or formula that gives the ..
possible values of x and the probability p(x) associated with each value.
Probability distributions can be used to describe the population
(3)
Shape
Outliers
Center and Spread (mean and standard deviation)
Let x be a discrete random variable with probability distribution p(x).
Then the mean, variance and standard deviation of x are given as
Mean: µ = Σx p(x)
Variance: σ2 = Σ(x-µ)2 p(x)
Standard Deviation: σ = √ σ2
