Réponse :
Bonjour,
Explications étape par étape :
1a.
[tex]\begin{array}{|c|c|c|c|c|}---&---&-----&-----&----\\&Gain&1er\ tirage&2eme\ tirage&proba\\NN&-1&0.4&0.4&0.16\\NB&*&0.4&0.6&0.24\\BN&*5&0.6&0.4&0.24\\BB&-9&0.6&0.6&0.36\\---&---&-----&-----&----\\\end {array}\\[/tex]
1b.
p(perdre 9€)=0.6*0.6=0.36
p(perdre 1€)=0.4*0.4=0.16
p(gagné 5€)=0.24+0.24=0.48
2.a
[tex]\begin{array}{|c|c|c|c|c|}---&---&-----&-----&----\\&Gain&1er\ tirage&2eme\ tirage&proba\\&&&&&NN&-1&\dfrac{N}{N+3} &\dfrac{N}{N+3} &(\dfrac{N}{N+3})^2\\&&&&&NB&*&\dfrac{N}{N+3}&\dfrac{3}{N+3}&\dfrac{3*N}{N+3}\\&&&&&BN&*5&\dfrac{3}{N+3}&\dfrac{N}{N+3}&\dfrac{3*N}{N+3}\\&&&&&BB&-9&\dfrac{3}{N+3}&\dfrac{3}{N+3}&\dfrac{9}{(N+3)^2}\\---&---&-----&-----&----\\\end {array}\\[/tex]
[tex]Gain=-9*\dfrac{9}{(N+3)^2}-1*\dfrac{N^2}{(N+3)^2}+5*\dfrac{6*N}{(N+3)^2}\\\\\\\boxed{Gain=\dfrac{-N^2+30N-81}{(N+3)^2}}\\[/tex]
2b.
[tex]-x^2+30x-81 > 0\\-(x-3)(x-27) > 0\\\\3 < x < 27\\[/tex]
2c.
Gain > 0 : 3 < N < 27
2d.
[tex](Gain)'=0\\\dfrac{(-2N+30)(N+3)^2-(-N^2+30N-81)*2*(N+3)}{(n+4)^4} =0\\\\(N+3)(-48N+432)=0\\\\N=9\ car\ N > 0[/tex]
