作者：Taylor Welch 8 年以前
327
Chapter 4: Probability-Formulas
The text delves into various fundamental concepts and rules of probability, including permutations and combinations. The Permutations Rule addresses scenarios where some items are identical, providing a formula to calculate the number of distinct permutations.
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Chapter 4: Probability-Formulas Combinations Rule "different combinations" or "order doesn't matter" # of combinations (order doesn't count)
when n diff. items are available, but only r of them
are selected w/o replacement nCr=n!/(n-r)!r! Permutations Rule (When some items are identical to others) 10 letters (aaaabbccde) are available and all
are selected, the # of diff. premutaions is:
10!/4!2!2! # of diff. permutations when n items
are available and all n are selected w/o
replacement, but some of the items are identical n!/(n_1 !n_2 !⋯n_k !) Permutations Rule (When all items are DIFFERENT) "If 5 letters are available, and
3 are to be selected w/o replacement" # of diff. permutations when n diff.
items are available, but only r of them
are selected w/o replacement nPr= n!
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(n-r)! Factorial Rule "the # of ways five letters can be arranged"
5!:5*4*3*2*1 n!: # of diff. permutations (order counts)
of n diff. items when all n of them are selected Factorial Counting Rule "How many different characters are possible
if they're all represented"
Ex: 00110111: 2^8=720 m*n=# of ways 2 events can occur Independent Occurrence doesn't affect probability:
Sampling w/ replacement Factorial symbol: ! Special: 0!=1 Example: 3!=3*2*1 Sampling w/o replacement Depedent Probability of event B occurring after
it is assumed that event A has already occurred P(A and B)= P(A)*P(B) (independent) P( A and B)=P(A)*P(B|A) (dependent) Intuitive Addition Rule Intuitive Addition Rule= P(A)+P(B)
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total # of outcomes in the sample space Rule of Complementary Events P(A)=1-P(Ā) P(Ā)=1-P(A) P(A)+P(Ā)=1 Formal Addition Rule P(A or B)= P(A) + P(B) - P(A and B) Probability of At Least One 4) Subtract the result from 1
P(at least 1 occurrence)= 1-P(no occurrences) 3) Find P(Ā) 2) Ā= getting none of the event 1) A= getting at least one of some event Conditional Probability P(B|A)= P(A and B)
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P(A) "given that..." Intuitive: P(B|A)
Equally likely outcomes P(A)= # of ways A can occur
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# of different simple events Subjective Probability P(A) is estimated by knowledge
of the relevant circumstances Relative Frequency P(A)= # of times that A occurred
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# of times trial was repeated Probability that event A DOESN'T occur P(Ā) Probability of event A P(A) Unusual: Extreme result # of outcomes is far above/below typical values Unlikely event Probability= 0.05 or less Multiplication And= multiplication P(A and B) Addition Or=addition P(A or B) Disjoint (Mutually Exclusive) Two events cannot occur at the same time
