Probability and Statistics
Basic Probability concepts
Most inspection and quality control theory deals with statistics to make inference about a population based on information contained in samples. The mechanism we use to make these inferences is probability.
We use P E to represent the probability of any event (E)
0
PE
1
The sum of all possible events = 1
P(S) = 1, S = Sample space
Definition of Probability
The ratio of the chances favoring an event to the total number of chances for and against the event. Probability
Is always a ratio.
Chances _ Favoring
P = Chances _ Favoring _ Plus _ Chances _ not _ Favoring
Empirical Probability
Empirical probabilities are nearly the only ones we know in industrial world. We watch and measure and count and calculate empirical probabilities from which we predict future probabilities. Also we stated that if an experiment is
Repeated a large number of times, (N), and the event (E) is observed nE times, the probability of E is approximately:
P(E)
nE
N
Simple Events
An event that cannot be decomposed is a simple event (E), or a sample point. The set of all sample points for an
Experiment is called sample space (S)
Compound Events
Compound events are formed by a composition of two or more events. They consist of more than one point in sample
Space. The two most important probability theorems are the additive and multiplicative laws. For the following
Discussion, EA = A and EB = B.
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I. Composition: A union or intersection of events
A. Union of A and B. A B
If A and B are two events in sample space (S), the union of A and B ( A B) contains all sample points in event A
Or B or both.
B. Intersection of A and B. ( A B ).
If A and B are two events in sample space (S), the intersection of A and B ( A B ) is composed of all sample
Points that are in both A and B.
II. Event relationships. There are three relationships in finding the probability of an event:
Complementary, conditional and mutually exclusive.
A. Complement of an event
The compliment of an event A is all sample points in the sample space (S), but not in A. The compliment of A
Is A , and P(A) 1 P(A)
B. Conditional probabilities
The conditional probability of event A given that B has occurred is:
P ( A | B)
P ( A B)
P (B )
If P(B) 0
Two events A and B are said to be independent if either:
P(A|B) = P(A) or P(B|A) = P(B)
Otherwise, the events are said to be dependent
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C. Mutually exclusive events
If event A contains no sample points in common with event B, then they are said to be mutually exclusive.
The Additive Law
If the two events are not mutually exclusive:
1. P A B P(A) P(B) P(A B)
If the two events are mutually exclusive, the law reduces to:
2. P A B P(A) P(B)
The Multiplicative Law
If events A and B are dependent, the probability of A influences the probability of B. This is known as conditional
Probability and the sample space is reduced.
1. P(A B) P(A) P(B | A) also
P(A
B)
P(B) P(A | B)
If events A and B are independent:
2. P(A B) P(A) P(B)
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Expected Value
Daniel Bernoulli stated that “expected value equals the sum of the values of each of a number of outcomes multiplied
By the probability of each outcome relative to all the other possibilities.”
If E represents the expected value operator and V represents the variance operator, such that:
V(X)
1.
4.
E[(X
)2 ]
2
If X is a random variable and c is a constant, then:
E(c) = c
V(c) = 0
2. E(X)=
5. V(X)=
2
3. E(cX)=cE(x)=c
6. V(cX)=c2 2
Permutations
An ordered arrangement of n distinct objects is called a permutation. The number of ways of ordering n distinct
Objects taken r at a time is designated by the symbol: Prn or P(n,r) and nPr
Counting rule for permutations
The number of ways that n distinct
