Korean, Edit

Chapter 2. Number of Cases

Higher category : 【Statistics】 Statistics Overview

1. overview  

2. sequence  

3. combination  

1. Overview

⑴ sampling

① with-replacement: re-injecting and extracting something already extracted

② without-replacement: extracting without re-injecting what has already been extracted.

⑵ types of number of cases

2. Permutation

⑴ definition : the number of cases in the order of k balls among n balls. consideration of order. without-replacement

⑵ permutation with repetition : consideration of order. with-replacement

⑶ permutation of multisets: permutation with something that are identical

⑷ circular permutation: number of cases sitting around a round table

3. Combination (binary coefficient) 

⑴ definition : number of cases of combination of k balls among n balls. no consideration of order. without-replacement

⑵ binary coefficient may be expressed in a triangule of Pascal

⑶ combination with repetition

① definition: consideration of order. with-replacement

② equivalent expression

a₁ + ··· + a_n = k, = k, a_i ≥ 0

⇔ A₁ + ··· + A_n = k+n, A_i ≥1

⇔ □ | □ | ··· | □,     # of □ = (k+n) and # of | = (k+n-1)

⇔ _nH_k = _n+k-1C_n-1 = _n+k-1C_k

⑷ formula 1.

⑸ formula 2.

⑹ formal 3.

① combinatoric interpretation

⑺ formula 4.

① algebraic interpretation

② combinatoric interpretation

○ _nC_k : number of cases in which k numbers are selected out of n numbers

○ _n-1C_k : number of cases in which k numbers are selected excluding a specific number.

○ _n-1C_k-1 : number of cases in which k numbers are selected including a specific number 

⑻ formula 5

① algebraic interpretation

② combinatoric interpretation 

○ situation: number of cases in which n+1 numbers are selected from 1 to n+m+1 numbers 

○ _nC_n : number of cases in which n+1 is the largest number of a combination

○ _n+1C_n : number of cases in which n+2 is the largest number of a combination 

○ _n+mC_n : number of cases in which n+m+1 is the largest number of a combination 

⑼ formula 6

① algebraic interpretation

② combinatoric interpretation

○ the probability of tossing a coin until the front or the back comes out n+1 times = 1

○ _nC_n × (½)ⁿ⁺¹ × 2 : the probability that the same side as the last side comes out n times, and the different side comes out 0 times 

○ _n+1C_n × (½)ⁿ⁺² × 2 : the probability that the same side as the last side will come out n times, and the different side will come out 1 time 

○ _2nC_n × (½)²ⁿ⁺¹ × 2 : the probability that the same side as the last side comes out n times, and the different side comes out n times

Input : 2019.06.27 09:48