Mathematics Dictionary

Mathematics Dictionary
Dr. K. G. Shih

Probability

Question
Four pair color balls. Put each two balls into four boxes

Symbol Defintion

Example : Sqr(x) = square root of x

Q01 | - Find probability all boxes are different color

Q02 | - Proposed formula

Q01. 4 pairs Color balls : Proposed formula

Quuestion

2 red, 2 green, 2 white and 2 blue balls
Pick 2 balls each time and put into 4 boxes

Solution

Balls : A, A, B, B, C, C, D, D
Sample space
- n(S) = C(8,2)*C(6,2)*C(4,2)*C(2,2)/(4*3*2*1)
- n(S) = (28*15*6*1)/(4*3*2*1)
- N(S) = 105
All box same color P = 1/105
One box and two boxes same color : S1 = C(6,2)*C(4,2)*C(2,2)/(3*2*1)
Two boxes same color : S2 = C(4,2)*C(2,2)/(2*1)
Three boxes same color : none
four boxes same color = C(2,2)/(1)
All box diff color n(0) = 1*n(S) - 4*S1 + 6*S2 - 4*S3 + 1
n(0) = 105 - 4*15 + 6*3 - 4*1 + 1 = 60
Probability for all boxes in diff color = P = n(0)/n(S) = 60/105
Coefficients 1, -4, +6, -4, + 1 are same binomial coefficients

Go to Begin

Q2. 4 pairs Color balls

Quuestion :

2 red, 2 green, 2 white and 2 blue balls
Balls : A, A, B, B, C, C, D, D
Pick 2 balls each time and put into 4 boxes
Find one box same color

Box 1 : C(8,2) = 28 samples.

4 same color and 24 different color
AA, AB, AB, AC, AC, AD, AD
AB, AB, AC, AC, AD, AD
BB, BC, BC, BD, BD
BC, BC, BD, BD
CC, CD, CD
CD, CD
DD
4 box same color : AA, BB, CC, DD
24 boxes different color

Box 1 same color AA and other 3 boxes BB, CC, DD

Based on 3 pairs results in Question P002

we have 6 three boxes all same
We have 36 only one box same color
We have 48 all boxes are diff color

Box 1 (A,A) combine with B,B,C,C,C,D,D
- (A,A) with other three boxes same color = 6 = all 4 boxes same color
- (A,A) with 36 one box same color = 36 = two boxes same color
- (A,A) 48 all boxes different color = 48 = one box same color
- Total samples for (A,A) (B,B) (C,C) (D,D)
  - 1 box same color 4*48 = 192
  - 2 boxes same color 4*36 = 144
  - 3 boxes same color = 0
  - 4 boxes same color = 4*6 = 24
Box 1 is (A,B) and box 2 is (A,B) and other two boxes C, C, D, D
- AB, AB, CC, DD
- AB, AB, CD, DC
- AB, AB, CD, DC
- AB, AB, CD, DC
- AB, AB, CD, DC
- AB, AB, DD, CC
- Samples
  - 1 box same color = 0
  - 2 boxes same = 2
  - 4 boxes different color = 4
- Box 1 = (A,B) box 2 = (A,B) : 2 (2 boxes same) and 4 (all boxes diff)
- Total
  - Two boxes same = 2*2 = 4
  - Four boxes same = 4*2 = 8
Box 1 is (A,B) and box 2 is not (A,B)
- Box 1 = (A,B) box 2 = (A,C) : 2 (1 box same) and 4 (all boxes diff)
- Box 1 = (A,B) box 2 = (A,C) : 2 (1 box same) and 4 (all boxes diff)
- Box 1 = (A,B) box 2 = (A,D) : 2 (1 box same) and 4 (all boxes diff)
- Box 1 = (A,B) box 2 = (A,D) : 2 (1 box same) and 4 (all boxes diff)
- Box 1 = (A,B) box 2 = (B,C) : 2 (1 box same) and 4 (all boxes diff)
- Box 1 = (A,B) box 2 = (B,C) : 2 (1 box same) and 4 (all boxes diff)
- Box 1 = (A,B) box 2 = (B,D) : 2 (1 box same) and 4 (all boxes diff)
- Box 1 = (A,B) box 2 = (B,D) : 2 (1 box same) and 4 (all boxes diff)
- Total
  - 1 boxes same color = 8*2 = 18
  - All different color boxes = 8*4 = 32
Box 1 = (A,B) and box 2 = (C,C) and four other balls A, B, D, D
- AB, CC, AB, DD
- AB, CC, AD, DA
- AB, CC, AD, DA
- AB, CC, BD, DB
- AB, CC, BD, DB
- AB, CC, DD, CC
- Box 1 = AB Box 2 = CC : 2 (2 same color boxes) and 4 (1 same color box)
- Box 1 = AB Box 2 = DD : 2 (2 same color boxes) and 4 (1 same color box)
- Total
  - 2 boxes same color = 2*2 = 4
  - 1 box same color = 2*4 = 8
Box 1 = AB and box 2 = CD and four other balls A, B, C, D
- AB, CD, AB, CD
- AB, CD, AC, BD
- AB, CD, AD, BC
- AB, CD, BC, AD
- AB, CD, BD, AC
- AB, CD, CD, AB
- Box 1 = AB box 2 = CD : All different color samples = 6
- Box 1 = AB box 2 = CD : All different color samples = 6
- Box 1 = AB box 2 = CD : All different color samples = 6
- Box 1 = AB box 2 = CD : All different color samples = 6
- Total
  - All boxes diff color = 4*6 = 24
Sumnary
- Total sample spaces n(S) = 28*15*6*1 = 2520
- All boxes diff color n(0) = 24*(4*6 + 1*4 + 8*4) = 24*60
- Two boxes same color n(2) = 144 + 24*2 + 24*4 = 288
- One box same color n(1)= 192 + 24*16 + 24*8 = 768
- All boxes same color n(4) = 24
Probability
- All boxes are in diff color : P = n(0)/n(S) = (24*60)/2520 = 60/105
- One box in same color : P = n(1) = 768/2520 = 32/105
- Two boxes in same color : P = n(2)/n(0) = 288/2520 = 12/105
- All boxes in same color : P = n(4)/n(0) = 024/2520 = 01/105
Go to Begin

AL 17 00. Outline

Go to Begin

Show Room of MD2002

Contact Dr. Shih

Math Examples Room