Les intervalles de fluctuation

Les paramètres

In [1]:
import math 
p=0.7
n=100

En seconde

In [2]:
k=1/math.sqrt(n)

a=p-k
b=p+k

print(f"I=[{a:.8f} ; {b:.8f}]")

I=[0.60000000 ; 0.80000000]

En terminale

In [3]:
k=1.96*math.sqrt(p*(1-p))/math.sqrt(n)

a=p-k
b=p+k

print(f"I=[{a:.8f} ; {b:.8f}]")

I=[0.61018152 ; 0.78981848]

En Première

In [4]:
#------------------------------------------------------
def fact(n) :
    "Calcule factoriel n."
    if (n==0) :
        return 1
    else :
        p=1
        for k in range(1,n+1):
            p=p*k
        return p

#------------------------------------------------------
def combin (n,k):
    "Calcule  k parmi n."
    p=1
    for i in range(n-k+1,n+1) :
        p=p*i
    return p//fact(k)



def binomial(n,p,k):
    "Calcule p(x=k)"
    return math.exp(math.log(combin(n,k))+k*math.log(p)+(n-k)*math.log(1-p))

cumul=0
d=dict() # dictionaire des (k,p(x=k))
for k in range(n+1):
    cumul+=binomial(n,p,k)
    d[k]=cumul

for (c,v) in d.items() : # recherche de a
    if v>0.025 :
        a=c
        break
        
for (c,v) in d.items() : # recherche de b
    if v>0.975 :
        b=c
        break

print(f"I=[{a/n:.8f} ; {b/n:.8f}]")

I=[0.61000000 ; 0.79000000]
In [16]:
d_bas=dict((c,v) for (c,v) in d.items() if a-10<c<a+10)
for (c,v) in d_bas.items():
    print(f"{c:4d} | {v:.6f}")
    if (c==a-1):
        print("------------------ 0,025")

  52 | 0.000117
  53 | 0.000256
  54 | 0.000537
  55 | 0.001086
  56 | 0.002115
  57 | 0.003968
  58 | 0.007174
  59 | 0.012498
  60 | 0.020989
------------------ 0,025
  61 | 0.033979
  62 | 0.053046
  63 | 0.079880
  64 | 0.116079
  65 | 0.162858
  66 | 0.220742
  67 | 0.289281
  68 | 0.366892
  69 | 0.450876
  70 | 0.537660
In [14]:
d_haut=dict((c,v) for (c,v) in d.items() if b-10<c<b+10)
for (c,v) in d_haut.items():
    print(f"{c:4d} | {v:.6f}")
    if (c==b-1):
        print("------------------ 0,975")

  70 | 0.537660
  71 | 0.623222
  72 | 0.703634
  73 | 0.775601
  74 | 0.836870
  75 | 0.886430
  76 | 0.924469
  77 | 0.952134
  78 | 0.971169
------------------ 0,975
  79 | 0.983537
  80 | 0.991113
  81 | 0.995477
  82 | 0.997837
  83 | 0.999031
  84 | 0.999595
  85 | 0.999843
  86 | 0.999943
  87 | 0.999981
  88 | 0.999994
In [ ]: