hw3 110590049

tags probability

2023PB_HW3.pdf

1.a

$$\frac{2^6{\text{C}}_6^{10}}{{\text{C}}_6^{20}}=0.3467$$

1.b

$$\frac{{\text{C}}_1^{10}{\text{C}}_4^9{2}^4}{{\text{C}}_6^{20}}=0.52$$

2

$$2^2(-1{)}^3{3}^2\times \frac{7!}{2!3!2!}=-7560$$

3

$${\text{C}}_3^9\times 2{\text{C}}_3^6=4620$$

4

$$\frac{{\text{C}}_2^6{5}^4}{6^6}=0.2$$

5

$$\begin{gathered} \left(\begin{array}{c}n\\ r\end{array}\right)={\text{C}}_r^n \\ {\text{C}}_r^{n+m}\text{mean combination of take ’r’ element from ’n+m’ element.} \\ \text{So is equal to all the combination that take ’k’ from ’n’ and take ’r-k’ of from ’m’ when r<n and r<m} \\ {\text{C}}_r^{n+m}=\sum _{k=0}^r{\text{C}}_k^n{\text{C}}_{r-k}^m \end{gathered}$$

6

$$1-\frac{{\text{C}}_5^{10}{\text{C}}_2^2}{{\text{C}}_7^{12}}=0.6818$$

7.a

$$\frac{4!{\text{C}}_4^8}{20!÷8!}=0.0144$$

7.b

$$1-\frac{({\text{C}}_3^{12}+{\text{C}}_1^8{\text{C}}_2^{12})}{{\text{C}}_3^{20}}=0.3428$$

8

$$\frac{{\text{C}}_3^{13}{\text{C}}_6^{39}9!}{{\text{C}}_9^{52}9!}\frac{{\text{C}}_1^{10}42!}{43!}=0.0589$$

9

$$0.4\ast 0.2+0.6\ast 0.16=0.176=17.6\%$$

10

$$\frac{1}{4}\times \frac{12}{51}+\frac{3}{4}\times \frac{13}{51}=0.25=25\%$$

11

$$\frac{{\text{C}}_3^{10}}{{\text{C}}_3^{18}}\frac{{\text{C}}_3^8}{{\text{C}}_3^{18}}+\frac{{\text{C}}_1^8{\text{C}}_2^{10}}{{\text{C}}_3^{18}}\frac{{\text{C}}_3^7}{{\text{C}}_3^{18}}+\frac{{\text{C}}_2^8{\text{C}}_1^{10}}{{\text{C}}_3^{18}}\frac{{\text{C}}_3^6}{{\text{C}}_3^{18}}+\frac{{\text{C}}_3^8}{{\text{C}}_3^{18}}\frac{{\text{C}}_3^5}{{\text{C}}_3^{18}}=0.038$$