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# For the events $A$ and $B$ , $P(A)=\dfrac{3}{4}$ , $P(B)=\dfrac{1}{5}$ , $P(A\cap B)=\dfrac{1}{20}$ then $P(A/B)=.....$.A.$\dfrac{1}{4}$B.$\dfrac{1}{15}$C.$\dfrac{3}{4}$D.$\dfrac{1}{2}$  Verified
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Hint: We have been given $P(A)=\dfrac{3}{4}$ , $P(B)=\dfrac{1}{5}$ , $P(A\cap B)=\dfrac{1}{20}$ and we have to find $P(A/B)=.....$ . So for that, use $P(A/B)=\dfrac{P(A\cap B)}{P(B)}$ . Try it, you will get the answer.

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true. Probability is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).

The literal meaning of Independent Events is the events which occur freely of each other. The events are independent of each other. In other words, the occurrence of one event does not affect the occurrence of the other. The probability of occurring of the two events are independent of each other.

An event A is said to be independent of another event B if the probability of occurrence of one of them is not affected by the occurrence of the other.

Suppose if we draw two cards from a pack of cards one after the other. The results of the two draws are independent if the cards are drawn with replacement i.e., the first card is put back into the pack before the second draw. If the cards are not replaced then the events of drawing the cards are not independent.

Statistically, an event $A$ is said to be independent of another event $B$, if the conditional probability$A$ of given $B$, i.e. $P(A|B)$ is equal to the unconditional probability of$A$.

The term mutually exclusive should not be mixed with the term independent. The term mutually exclusive is related to the occurrence of an event. By independence of events, we mean the independence of probability of occurrence of events.
We have to find $P(A/B)$ .

We are given that $P(A)=\dfrac{3}{4}$ , $P(B)=\dfrac{1}{5}$ , $P(A\cap B)=\dfrac{1}{20}$ .
So we know, $P(A/B)=\dfrac{P(A\cap B)}{P(B)}$ .
So substituting the values we get,
$P(A/B)=\dfrac{\dfrac{1}{20}}{\dfrac{1}{5}}=\dfrac{1}{4}$
So we get $P(A/B)=\dfrac{1}{4}$ .
The correct answer is option (A).

Note: Read the question and see what is asked. Your concept regarding probability should be clear. A proper assumption should be made. Do not make silly mistakes while substituting. Equate it in a proper manner and don't confuse yourself.