21 P. 128

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1. $P_A(B) = \dfrac{P(A\cap B)}{P(A)} = \dfrac{7/15}{3/5} =\dfrac{7}{15}\times\dfrac{5}{3} = \dfrac{7}{9}$.

2. Arbre pondéré :

3. $P_{\overline A}(B) = 1 - P_{\overline A}(\overline B) = 1 - \dfrac 2 7 = \dfrac 5 7$.

4.Calculons : \[\begin{aligned} P(B) &= P(A\cap B) + P(\overline A \cap \overline B)& \\ &=P(A\cap B) + P(\overline A)\times P_{\overline A}(B)& \\ &= \frac 7{15} + \frac 25 \times \frac 5 7& \\ &=\frac 7{15} + \frac 2 7& \\ &=\frac{79}{105}.& \end{aligned}\]

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code : 128021