每个球都被抽过时抽取次数期望为无穷，设有n个球，有放回的随机抽m次，m次一共抽到不同球种类的期望是

当m为正无穷时E收敛于n

假设n为1000，m为10000，使用python简单画个图

num = 1000

i_l = []

a_l = []

p_l = []

for i in range(2, 10002):

a = 0

for k in range(1, i):

a = a + num / (num - 1) * (((num - 1) / num) ** k)

p = a / num

i_l.append(i)

a_l.append(a)

p_l.append(p)

if (i - 1) % 500 == 0:

print(i - 1, a, p)

plt.figure(figsize=(20,10))

plt.subplot(121)

plt.scatter(i_l, i_l, marker = 'o', color = 'r', label='retry frequency', s = 5)

plt.legend(loc = 'lower right')

plt.title('retry frequency')

plt.subplot(122)

plt.scatter(i_l, p_l, marker = 'o', color = 'r', label='retry frequency', s = 5)

plt.legend(loc = 'lower right')

plt.title('success probability')

plt.show()

500 393.62105513881454 0.3936210551388145

1000 632.3045752290353 0.6323045752290353

1500 777.0372362970971 0.7770372362970971

2000 864.8000746024981 0.8648000746024981

2500 918.0176118921498 0.9180176118921498

3000 950.2876060019588 0.9502876060019588

3500 969.8554509809422 0.9698554509809423

4000 981.7209801725057 0.9817209801725058

4500 988.9159872439054 0.9889159872439054

5000 993.2788880401297 0.9932788880401296

5500 995.9244592214772 0.9959244592214771

6000 997.5286778829792 0.9975286778829792

6500 998.5014423022661 0.9985014423022661

7000 999.0913061644318 0.9990913061644318

7500 999.4489871907842 0.9994489871907842

8000 999.6658774341404 0.9996658774341404

8500 999.7973951110588 0.9997973951110588

9000 999.8771446612194 0.9998771446612194

9500 999.9255031092993 0.9999255031092993

10000 999.9548266540206 0.9999548266540206

10000 999.9548266540206 0.9999548266540206