Document annexe sur les probabilités - IREM Clermont

n
p < 0,5p≈
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•
n p
•X
? X 0, . . . , n
•k k 0n
?P(X=k) = n
kpk(1 −p)n−k
?n
k=n! k!(n−k)!
p= 0,01 n= 3
•k
?P(X=k) = 3
k0,01k(0,99)3−k
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?P(X= 0) = 0,993≈97,030%
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?P(X= 1) = 3 ×0,01 ×0,992≈2,940%
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?P(X= 2) = 3×2
2×0,012×0,99 ≈0,030%
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?P(X= 3) ≈0,0001%
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?1− P(X= 0) ≈2,97010%
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?P(X= 1) + P(X= 2) ≈2,97000%
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?≈99,997%
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?P(X= 1) + P(X= 2) ≈2,97%
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?P(X= 1) ≈2,94%
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?≈99%

p= 0,001 n= 36
•k
?P(X=k) = 36
k0,001k(0,999)36−k
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?P(X= 0) = 0,99936 ≈96,462%
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?1− P(X= 0) ≈3,54%
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?P(X= 1) = 36 ×0,001 ×0,99935 ≈3,48%
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?≈98%
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?1− P(X= 0) − P(X= 1) ≈0,06%
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?≈1,7%