概率论第一章作业

5

(1)

${P(A)=\frac{C_{5}^{2}C_{5}^{3}+C_{5}^{3}C_{5}^{2}+C_{5}^{4}C_{5}^{1}+C_{5}^{5}}{C_{10}^{5}}=1-P(\overline A)}=1-\frac{C_{5}^{1}C_{5}^{4}+C_{5}^{5}}{C_{10}^{5}}=\frac{226}{252}=\frac{113}{126}$

t=choose(5,2)*choose(5,3)+choose(5,3)*choose(5,2)+choose(5,4)*choose(5,1)+choose(5,5)
s=choose(10,5)
p=t/s
p

(2)

${P(ABC)=P(C|AB)P(B|A)P(A)}=\frac{1}{2}*\frac{4}{9}*\frac{3}{8}=\frac{1}{12}$

> pA=1/2
> pB=4/9
> pC=3/8
> p=pA*pB*pC
> p
[1] 0.08333333

6

(1)

${P(A)=\frac{C_{5}^{2}}{C_{10}^{3}}}=\frac{1}{12}$

> choose(5,2)/choose(10,3)
[1] 0.08333333

(2)

${P(A)=\frac{C_{4}^{2}}{C_{10}^{3}}}=\frac{1}{20}$

> choose(4,2)/choose(10,3)
[1] 0.05

7

${P(A)=\frac{C_{10}^{4}C_{4}^{3}C_{3}^{2}}{C_{17}^{9}}=\frac{252}{2431}}$

> choose(10,4)*choose(4,3)*choose(3,2)/choose(17,9)
[1] 0.103661

8

(1)

${P(A)=\frac{C_{400}^{90}C_{1100}^{110}}{C_{1500}^{200}}}$

> choose(400,90)*choose(1100,110)/choose(1500,200)
[1] 8.234075e-10

(2)

${P(A)=1-P(\overline A)=1-\frac{C_{400}^{1}C_{1100}^{199}+C_{1100}^{200}}{C_{1500}^{200}}}$

> 1-(choose(400,1)*choose(1100,199)+choose(1100,200))/choose(1500,200)
[1] 1

9

${P(A)=1-P(\overline A)=1-\frac{C_{5}^{4}{C_{2}^{1}}^4}{C_{10}^{4}}}=\frac{13}{21}$

> 1-choose(5,4)*choose(2,1)^4/choose(10,4)
[1] 0.6190476

10

${P(A)=\frac{C_{1}^{1}C_{2}^{1}C_{2}^{1}C_{1}^{1}C_{1}^{1}C_{1}^{1}C_{1}^{1}}{A_{11}^{7}}}=\frac{1}{41580}$

> choose(2,1)*choose(2,1)/(choose(11,7)*factorial(7))
[1] 2.405002e-06

11

${P(A)=\frac{C_{4}^{3}}{4^3}}$

${P(B)=\frac{C_{4}^{1}C_{3}^{2}}{4^3}}$

${P(C)=\frac{C_{4}^{1}C_{3}^{1}}{4^3}}$

12

${P(A)=\frac{C_{10}^{1}C_{47}^{3}...C_{26}^{3}C_{23}^{3}}{C_{50}^{3}C_{47}^{3}...C_{26}^{3}C_{23}^{3}}=\frac{C_{10}^{1}}{C_{50}^{3}}}=\frac{1}{1960}$

> pA=choose(4,3)/4^3
> pB=choose(4,1)*choose(3,2)/4^3
> pC=choose(4,1)*choose(3,1)/4^3
> pA
[1] 0.0625
> pB
[1] 0.1875
> pC
[1] 0.1875

13

(1)

${P(A)=\frac{C_{5}^{1}C_{2}^{1}C_{3}^{1}C_{2}^{1}}{C_{12}^{5}}=\frac{4}{33}}$

> 5*2*3*2/choose(12,5)
[1] 0.07575758

(2)

${P(A)=\frac{C_{5}^{2}C_{2}^{1}C_{3}^{1}C_{2}^{1}+C_{5}^{1}C_{2}^{2}C_{3}^{1}C_{2}^{1}+C_{5}^{1}C_{2}^{1}C_{3}^{2}C_{2}^{1}+C_{5}^{1}C_{2}^{1}C_{3}^{1}C_{2}^{2}}{C_{12}^{5}}}=\frac{10}{33}$

> (choose(5,2)*choose(2,1)*choose(3,1)*choose(2,1)+choose(5,1)*choose(2,2)*choose(3,1)*choose(2,1)+choose(5,1)*choose(2,1)*choose(3,2)*choose(2,1)+choose(5,1)*choose(2,1)*choose(3,1)*choose(2,2))/choose(12,5)
[1] 0.3030303

f = function(x)
  sqrt(2)
x = seq(1.5, 3.5, length=100)
y = rep(0, length(x))
j = 1
for (i in x) {
  y[j] = f(i)
  j = j + 1
}
plot(x, y, type='o')
abline(v = 2)
abline(v = 3)
s = 100
a = 0
cnt <- 1
while (cnt <= 100000) {
    xx = 2 + runif(1)
    yy = 100 * runif(1)
    if( f(xx) > yy ) {
        a = a + 1
        points(x = xx, y = yy, pch = 20, cex = 1, col = "red")
    }
    else
        points(x = xx, y = yy, pch = 20, cex = 1, col = "blue")
  cnt = cnt + 1
}
print((a / 10000) * s)