# Suppose that A and B are independent events such that P(A)=0.10 and P(\bar{B})=0.20 Find P(A\cap B) and P(A\cup B)

Suppose that A and B are independent events such that $$P(A)=0.10$$ and $$P(\bar{B})=0.20$$
Find $$P(A\cap B)$$ and $$P(A\cup B)$$

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Step 1
The value of $$P(A\cap B)$$ is obtained below:
From the given information, the events A and B are independent events and the probability values are
$$P(A)=0.10$$, and the value of $$P(B)$$ is,
$$P(B)=1-P(\bar{B})$$
$$=(1-0.20)$$
$$P(B)=0.80$$
The required probability is,
$$P(A\cap B)=P(A)\times P(B)$$
$$=0.10\times0.80$$
$$=0.08$$
The value of $$P(A\cap B)$$ is $$0.08$$
The value of $$P(A\cap B)$$ is obtained by taking the product of probability of event A and the probability of event B. It can be expected that $$2\%$$ of the event A and B occurs.
Step 2
The value of $$P(A\cup B)$$ is obtained as shown below:
The probability is,
$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$
$$=0.10+0.80-0.08$$
$$=0.90-0.08$$
$$=0.82$$
The value of $$P(A\cup B)$$ is $$0.82$$.
The value of $$P(A\cup B)$$ is obtained by adding the individual probabilities and then subtracting the probability of A and B to the resulted value. It can be expected that about $$82\%$$ of times event A or B happens.
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