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Fill in the blanks:
$P(A)$$P(B)$$P(A \cap B)$$P(A \cup B)$
$(i)$$\dfrac{1}{3}$$\dfrac{1}{5}$$\dfrac{1}{{15}}$
$(ii)$$0.35$$0.25$$0.6$
$(iii)$$0.5$$0.35$$0.7$

Answer
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Hint: In the above question we have to find some unknown values and some probability values are known to us. Use the basic formulae of \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] to find out the value of these unknown quantities this will help to reach the right values of these quantities.

$(i)$ In the first part, we are given the values as
$P(A) = \dfrac{1}{3}$, $P(B) = \dfrac{1}{5}$,$P(A \cap B) = \dfrac{1}{{15}}$
We know that
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
Therefore, by using this formula and substituting the values of equation (1) in equation (2) we get
$P(A \cap B) = \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{{15}}$
$ = \dfrac{{5 + 3 - 1}}{{15}}$
$ = \dfrac{{8 - 1}}{{15}}$
$ = \dfrac{7}{{15}}$
$\therefore P(A \cap B) = \dfrac{7}{{15}}$

$(ii)$ In this part, we are given the values as
$P(A) = 0.35$,$P(A \cap B) = 0.25$,$P(A \cup B) = 0.6$ … (1)
We know that,
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] … (2)
Therefore, by using this formula and substituting the values of equation (1) in equation (2) we get
\[0.6 = 0.35 + P(B) - 0.25\]
\[P(B) = 0.6 - 0.35 + 0.25\]
\[P(B) = 0.6 - 0.10\]
\[\therefore P(B) = 0.5\]

$(iii)$ In this part, we are given the values as
$P(A) = 0.5$, $P(B) = 0.35$,$P(A \cup B) = 0.7$
We know that,
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
Therefore, by using this formula and substituting the values of equation (1) in equation (2) we get
\[0.7 = 0.5 + 0.35 - P(A \cap B)\]
\[P(A \cap B) = 0.5 + 0.35 - 0.7\]
\[P(A \cap B) = 0.85 - 0.7\]
\[\therefore P(A \cap B) = 0.15\]

Therefore, after getting the required solutions, we can fill the table in the form:

$P(A)$$P(B)$$P(A \cap B)$$P(A \cup B)$
$(i)$$\dfrac{1}{3}$$\dfrac{1}{5}$$\dfrac{1}{{15}}$$\dfrac{7}{{15}}$
$(ii)$$0.35$$0.5$$0.25$$0.6$
$(iii)$$0.5$$0.35$$0.15$$0.7$


Note: Whenever we face such types of problems the key point is to have a good grasp of the probability formula, some of them are mentioned above while performing the solution. Let’s talk about the physical interpretation of $A \cup B$ this means that we have to find the probability of occurrence of event A and event B. This same concept can be applied to get the physical interpretation of $A \cap B$ as well.