Download Chapter 4-Formula Sheet

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Probability Concepts (Chapter 4)
Range of probabilities rule
The probability of an event E is between 0 and 1, inclusive. That is, 0 ≤ P(E) ≤ 1
The probability of a certain event is 1. That is, P(E) = 1
The probability of a certain event is 0. That is, P(E) = 0
Type of Probably problems
1) Relative Frequency Approximation of Probability & Classical Probability
P( A) =
Frequency of Event A
Sample Size
P( A) =
Favorable Outcomes
Number of Outcomes in the Sample Space
2) Addition Rule: The probability that event A or B will occur is given by
P (A or B) = P (A) + P (B) – P (A and B ).
If events A and B are mutually exclusive, then the rule can be simplified to
P (A or B) = P (A) + P (B)
Mutually exclusive: Two events A and B cannot occur at the same time
3) The Multiplication rule: We use the multiplication rule when we find the probability of two (or more) events, “A and B.”
( A) P( B | A) where P( B | A) B given A has occured .
Dependent
=
Events : P (A and B) P=
Indepen det Events: P (A and B) = P (A) P ( B )
4) Conditional Probability: The probability of an event occurring, given that another event has already occurred. We denoted P(B | A) (read
“probability of B, given A”)
P(B | A) =
P(A and B)
P( A)
Useful Rules:
• P (At least one) = 1 − P (none)
•
Let A = Event A, then Complement of A = A = Not A . We find the complement by P (A) = 1 − P (A)
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Counting
Counting Rules:
1) Fundamental Counting Rule: If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can
occur in sequence is m•n. We can be extended for any number of events occurring in sequence.
•
Some of the counting rules we encounter involve the factorial symbol.
For any integer n ≥ 0, the factorial symbol n! is defined as follows:
0! = 1
1! = 1
n! = n(n - 1) (n - 2) (n - 3)…3 • 2 • 1
2) Permutations: A permutation is an arrangement of items, such that:
• r items are chosen at a time from n distinct items
• Repetition of items is not allowed
• The order of the items is important
The number of permutations of n items chosen r at a time is denoted as nPr, and given by the formula
n
Pr =
n!
(n − r )!
3) Distinguishable Permutations: The number of distinguishable permutations of n objects where n1 are of one type, n2 are of another type, and so on
n!
n1 !⋅ n2 !⋅ n3 !⋅⋅⋅ nk !
where n1 + n2 + n3 +···+ nk = n
4) Combinations: A combination is an arrangement of items in which:
• r items are chosen at a time from n distinct items
• repetition of items is not allowed
• the order of the items is not important
The number of combinations of r items chosen from n items is denoted as nCr, and given by the formula
n
Cr =
n!
(n − r )!r !