一组人群中出现重复生日的概率是多少?
忽然想起这个命题, 在若干年以前, 军训的时候教官说我们班上40多人里必然有两个人同一天过生日, 还说没看过我们的生日, 感觉跟魔术一样. 让大家验证了一下, 居然有两组人的生日都在同一天. 而在之前公司里面20多个人, 也有同一天过生日的. 那么究竟一组人群里重复生日的概率是多少呢? 想验证一下.
用node.js来验证, 验证10万组:
const testTimes = 100000
const yearLength = 365
function main(){
var doubledBirthdaysArrayLengths = []
for(let i = 0; i < testTimes; i++){
doubledBirthdaysArrayLengths.push(doubledLength())
}
console.log("一组人群中有相同生日的概率是多少?")
console.log("有重复生日的人群的人数: ")
console.log(doubledBirthdaysArrayLengths)
console.log('生日重复的一组中最多人数: ', Math.max(...doubledBirthdaysArrayLengths))
console.log('生日重复的一组中最少人数: ', Math.min(...doubledBirthdaysArrayLengths))
console.log('生日重复的一组中平均人数: ', average(doubledBirthdaysArrayLengths))
distribute(doubledBirthdaysArrayLengths)
}
function distribute(arr){
let peopleNumsofGroups = [] //一组中的人数为数组下标, 数组值为此组人数出现的次数.
for(let i = 0; i < yearLength; i+=1){
peopleNumsofGroups.push(0)
}
for(let a of arr){
peopleNumsofGroups[a]++
}
// console.log('Distribute: ', days)
console.log('重复生日人群, 某一人数出现的次数:')
printArrayInTable(peopleNumsofGroups, 0)
let probes = [], sum = average(peopleNumsofGroups) * peopleNumsofGroups.length
for(let i = 0; i < yearLength; i+=1){
if(i == 0){
probes[i] = peopleNumsofGroups[i]/sum * 100
}else{
probes[i] = peopleNumsofGroups[i]/sum * 100 + probes[i - 1]
}
}
console.log('概率:')
printArrayInTable(probes, 2)
}
function printArrayInTable(arr, fixDigits){
console.log('\t[0]\t[1]\t[2]\t[3]\t[4]\t[5]\t[6]\t[7]\t[8]\t[9]')
for(let row = 0; row < yearLength / 10 + 1; row+=1){
let out = '[' + row + ']\t'
for(let col = 0; col < 10; col+=1){
let index = row*10 + col
if(index >= yearLength){
return
}
out += arr[row*10 + col].toFixed(fixDigits) + '\t'
}
console.log(out)
}
}
function average(arr){
let sum = 0
for(let a of arr){
sum += a
}
return sum / arr.length
}
function doubledLength(){
let birthdays = []
for(let i = 0; i < yearLength + 1; i+=1){
var birthday = parseInt(Math.random()*yearLength)
if(findDouble(birthday, birthdays)){
birthdays.push(birthday)
// console.log("birthdays: ", birthdays)
return birthdays.length
}
birthdays.push(birthday)
}
return birthdays.length
}
function findDouble(obj, arr){
for(let a of arr){
if(a == obj){
return true
}
}
return false
}
main()
输出结果如下:
一组人群中有相同生日的概率是多少?
有重复生日的人群的人数:
[
19, 22, 14, 3, 13, 25, 10, 24, 56, 5, 15, 27,
25, 18, 27, 19, 22, 38, 21, 41, 9, 13, 37, 36,
28, 25, 22, 29, 11, 23, 19, 14, 13, 16, 32, 26,
8, 19, 7, 36, 6, 22, 18, 23, 20, 33, 32, 21,
64, 22, 48, 43, 5, 16, 14, 7, 19, 32, 15, 16,
15, 33, 58, 19, 26, 18, 45, 31, 4, 5, 30, 26,
14, 19, 31, 14, 6, 9, 27, 12, 47, 51, 7, 21,
16, 42, 66, 15, 28, 32, 18, 10, 12, 14, 10, 16,
23, 15, 23, 8,
... 99900 more items
]
生日重复的一组中最多人数: 84
生日重复的一组中最少人数: 2
生日重复的一组中平均人数: 24.60991
重复生日人群, 某一人数出现的次数:
[0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
[0] 0 0 279 577 842 1006 1334 1520 1873 1949
[1] 2223 2441 2627 2684 2975 2878 3110 3148 3229 3227
[2] 3269 3199 3180 3195 2974 3076 3014 2846 2747 2650
[3] 2624 2440 2327 2111 2096 1863 1807 1647 1564 1399
[4] 1271 1162 1023 996 884 773 738 642 575 541
[5] 460 394 384 325 277 247 203 173 143 148
[6] 111 97 79 71 55 53 41 36 31 18
[7] 24 16 9 16 4 5 7 2 6 6
[8] 1 0 1 0 2 0 0 0 0 0
[9] 0 0 0 0 0 0 0 0 0 0
[10] 0 0 0 0 0 0 0 0 0 0
[11] 0 0 0 0 0 0 0 0 0 0
[12] 0 0 0 0 0 0 0 0 0 0
[13] 0 0 0 0 0 0 0 0 0 0
[14] 0 0 0 0 0 0 0 0 0 0
[15] 0 0 0 0 0 0 0 0 0 0
[16] 0 0 0 0 0 0 0 0 0 0
[17] 0 0 0 0 0 0 0 0 0 0
[18] 0 0 0 0 0 0 0 0 0 0
[19] 0 0 0 0 0 0 0 0 0 0
[20] 0 0 0 0 0 0 0 0 0 0
[21] 0 0 0 0 0 0 0 0 0 0
[22] 0 0 0 0 0 0 0 0 0 0
[23] 0 0 0 0 0 0 0 0 0 0
[24] 0 0 0 0 0 0 0 0 0 0
[25] 0 0 0 0 0 0 0 0 0 0
[26] 0 0 0 0 0 0 0 0 0 0
[27] 0 0 0 0 0 0 0 0 0 0
[28] 0 0 0 0 0 0 0 0 0 0
[29] 0 0 0 0 0 0 0 0 0 0
[30] 0 0 0 0 0 0 0 0 0 0
[31] 0 0 0 0 0 0 0 0 0 0
[32] 0 0 0 0 0 0 0 0 0 0
[33] 0 0 0 0 0 0 0 0 0 0
[34] 0 0 0 0 0 0 0 0 0 0
[35] 0 0 0 0 0 0 0 0 0 0
概率:
[0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
[0] 0.00 0.00 0.28 0.86 1.70 2.70 4.04 5.56 7.43 9.38
[1] 11.60 14.04 16.67 19.36 22.33 25.21 28.32 31.47 34.70 37.92
[2] 41.19 44.39 47.57 50.77 53.74 56.81 59.83 62.67 65.42 68.07
[3] 70.70 73.14 75.46 77.57 79.67 81.53 83.34 84.99 86.55 87.95
[4] 89.22 90.38 91.41 92.40 93.29 94.06 94.80 95.44 96.01 96.56
[5] 97.02 97.41 97.79 98.12 98.40 98.64 98.85 99.02 99.16 99.31
[6] 99.42 99.52 99.60 99.67 99.72 99.78 99.82 99.85 99.88 99.90
[7] 99.93 99.94 99.95 99.97 99.97 99.98 99.98 99.98 99.99 100.00
[8] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[9] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[10] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[11] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[12] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[13] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[14] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[15] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[16] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[17] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[18] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[19] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[20] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[21] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[22] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[23] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[24] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[25] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[26] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[27] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[28] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[29] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[30] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[31] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[32] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[33] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[34] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00
[35] 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00可以看到, 班上有40个人的时候, 有重复生日的概率已经高达90%左右了. 按中国班级的规定, 小学标准是45人,有94%的概率有两人同一天过生日. 初高中是50人, 概率则高达97%.如果把两个班放在一起, 概率就是100%了.