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If $$P(A)=0.3, P(B)=0.2,$$ and $$P(A \cap B)=0.1$$

$$P(A)=0.3$$
$$P(B)=0.2$$
$$P(A \cap B)=0.1$$

$$P\left(A^{\prime}\right)=1-P(A)=1-0.3=0.7$$

b) $$P(A \cup B)=P(A)+P(B)-P(A \cap B)$$

$$=0.3+0.2-0.1=0.4$$

c) $$P\left(A^{\prime} \cap B\right)$$

$$P(B)=P(A \cap B)+P(A^{\prime} \cap B)$$

$$0.2=0.1+P(A^{\prime} \cap B)$$ 

$$\Rightarrow P(A^{\prime} \cap B) = 0.1$$

d) $$P\left(A \cap B^{\prime}\right)$$

$$P(A)=P(A \cap B)+P(A \cap B^{\prime})$$

$$0.3=0.1+P(A \cap B^{\prime})$$

$$P(A \cap B^{\prime})=0.2$$

e) $$P\left[(A \cup B)^{\prime}\right]$$

$$=1-P(A \cup B)=0.6$$

f) $$P\left(A^{\prime} \cup B\right)$$

$$=P\left(A^{\prime}\right)+P(B)-P\left(A^{\prime} \cap B\right)$$

$$=0.7+0.2-0.1=0.8$$

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