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$$\text { If } P(A | B)=0.3, P(B)=0.8, \text { and } P(A)=0.3, \text { are }$$
the events $$B$$ and the complement of $$A$$ independent?

$$P(A | B)=0.3$$
$$P(B)=0.8$$
$$P(A)=0.3$$

$$B, A^\prime$$ independed??

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

$$P\left(A^{\prime} / B\right)=1-P(A | B)$$

$$=1-0.3=0.7$$

$$P\left(A^{\prime} | B\right)=P\left(A^{\prime}\right) \Rightarrow A^{\prime}, B \rightarrow$$ independent

A batch of 500 containers for frozen orange juice con-
tains 5 that are defective. Two are selected, at random, without
replacement, from the batch. Let $$A$$ and $$B$$ denote the events
that the first and second container selected is defective, re-
spectively.

(a) Are $$A$$ and $$B$$ independent events?
(b) If the sampling were done with replacement, would $$A$$ and
$$B$$ be independent?

batch $$\rightarrow 500$$

5 defective

495

$$A \rightarrow$$ first is defective
$$B \rightarrow$$ second is defective

a) $$P(A | B)=P(A)$$
$$P(B | A)=P(B)$$

$$P(A)=\frac{5}{500}=\frac{1}{100} $$

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

$$=\frac{4}{499} \times \frac{5}{500}+\frac{5}{499} \times \frac{495}{500}=\frac{5}{500}=\frac{1}{100} $$

$$P(B | A) \neq P(B)$$

$$\text { Not independet } $$

b) $$\rightarrow A, B \rightarrow$$ independet

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