hw3 1.a 1.b 2 3 4 5 6 7.a 7.b 8 9 10 11

hw3

tags probability

2023PB_HW3.pdf

1.a

26C610C620=0.3467\frac{2^6\text{C}_6^{10}}{\text{C}_6^{20}}=0.3467

1.b

C110C4924C620=0.52\frac{\text{C}_1^{10}\text{C}_4^{9}2^{4}}{\text{C}_6^{20}}=0.52

2

22(−1)332×7!2!3!2!=−75602^2(-1)^33^2\times\frac{7!}{2!3!2!}=-7560

3

C39×2C36=4620\text{C}_3^{9}\times2\text{C}_3^{6}=4620

4

C265466=0.2\frac{\text{C}_2^{6}5^4}{6^6}=0.2

5

(nr)=CrnCrn+mmean combination of take ’r’ element from ’n+m’ element.So is equal to all the combination that take ’k’ from ’n’ and take ’r-k’ of from ’m’ when r<n and r<mCrn+m=∑k=0rCknCr−km\begin{pmatrix} n\\ r \end{pmatrix} =\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}

6

1−C510C22C712=0.68181-\frac{\text{C}_{5}^{10}\text{C}_{2}^{2}}{\text{C}_7^{12}} =0.6818

7.a

4!C4820!÷8!=0.0144\frac{4!\text{C}_4^8}{20!\div8!}=0.0144

7.b

1−(C312+C18C212)C320=0.34281-\frac{(\text{C}_3^{12}+\text{C}_1^8\text{C}_2^{12})}{\text{C}_3^{20}}=0.3428

8

C313C6399!C9529!C11042!43!=0.0589\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∗0.2+0.6∗0.16=0.176=17.6%0.4*0.2+0.6*0.16=0.176=17.6\%

10

14×1251+34×1351=0.25=25%\frac{1}{4}\times\frac{12}{51}+\frac{3}{4}\times\frac{13}{51}=0.25=25\%

11

C310C318C38C318+C18C210C318C37C318+C28C110C318C36C318+C38C318C35C318=0.038\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