Question

Let A and B be two events such that $p(A) = 0.3A$ and $p(A \cup B) = 0.8$ . If A and B are independent events, then P(B) =
1) $\dfrac{5}{6}$
2) $\dfrac{5}{7}$
3) $\dfrac{3}{5}$
4) $\dfrac{2}{5}$

Answer 1

Hint: In order to solve this question, you have to know the exact option of the event A and B are independent if the equation $p(A \cup B) = p(A) \times p(B)$ hold true. You can use the equation to check if events are independent, multiply the probabilities of the two events together to see if they equal the probability of them both happening together.

Complete answer:
As A and B are independent events,
$p(A \cup B) = 0.8$ is given in the question, and $p(A) = 0.3A$ is also given above.
Without the knowledge of the occurrence of B, information about the occurrence of A would simply be $p(A)$ The probability of A knowing that event B has or will have occurred, will be the probability of relative to $p(B)$, the probability that B has occurred.
$ \Rightarrow p(A \cup B) = p(A) + p(B) - p(A)$ .
A conditional probability can always be computed using the formula in the definition.
Sometimes it can be computed by discarding part of the sample space. Two events A and B are independent if the probability$p(A \cap B)$ of their intersection A∩B is equal to the product $p(A) \times p(B)$ of their individual probabilities

Let’s assemble the events value correctly,
$ \Rightarrow 0.8 = 0.3 + p(B) - 0.3p(B)$
$ \Rightarrow 0.8 - 0.3 = p(B)(1 - 0.3)$
$ \Rightarrow 0.5 = p(B)(0.7)$
Multiply by 10 on both sides and we will get,
$ \Rightarrow 5 = p(B)7$
Then,
$ \Rightarrow p(B) = \dfrac{5}{7}$
So, Finally correct answer is
$ \Rightarrow p(B) = \dfrac{5}{7}$

Note:
Two events are independent if the result of the second event is not affected by the first event. If A and B are independent events, the probabilities of both events occurring is the product of the individual events. Important to distinguish independence from mutually exclusive which would say $(B \cap A)$ is empty(cannot happen).