Question

If events A and B are independent, and \[P\left( A \right) = 0.15\], \[P\left( {A \cup B} \right) = 0.45\], then \[P\left( B \right) = \]
(a) \[\dfrac{6}{{13}}\]
(b) \[\dfrac{6}{{17}}\]
(c) \[0.315\]
(d) \[0.352\]

Answer 1

Hint:
Here, we need to find the probability of event B. We will use the probability of intersection of independent events to find the intersection between two events. Then we will use the formula for the probability of union of two events and find the probability of \[P\left( B \right)\].

Formula Used:
We will use the following formulas:
1) The probability of the intersection of two independent events is equal to the product of their individual probabilities. This can be written as \[P\left( {A \cap B} \right) = P\left( A \right) \times P\left( B \right)\].
2) The probability of the union of two sets can be found using the formula \[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\].

Complete step by step solution:
We will use the probability of intersection of independent events, and the formula for the probability of union of two events to find the probability of \[P\left( B \right)\].
Let \[P\left( B \right) = x\].
It is given that the two events A and B are independent events.
Substituting \[P\left( A \right) = 0.15\] and \[P\left( B \right) = x\] in the formula \[P\left( {A \cap B} \right) = P\left( A \right) \times P\left( B \right)\], we get
\[ \Rightarrow P\left( {A \cap B} \right) = 0.15 \times x\]
Thus, we get
\[ \Rightarrow P\left( {A \cap B} \right) = 0.15x\]
Now, it is given that the probability of the union of the two events A and B is \[P\left( {A \cup B} \right) = 0.45\].
Substituting \[P\left( {A \cup B} \right) = 0.45\], \[P\left( A \right) = 0.15\], \[P\left( B \right) = x\], and \[P\left( {A \cap B} \right) = 0.15x\] in the formula \[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\], we get
\[ \Rightarrow 0.45 = 0.15 + x - 0.15x\]
This is a linear equation in terms of \[x\]. We will solve this equation to find the value of \[x\] and hence, the probability of event B.
Subtracting the like terms, we get
\[ \Rightarrow 0.45 = 0.15 + 0.85x\]
Subtracting \[0.15\] from both sides, we get
\[ \Rightarrow 0.30 = 0.85x\]
Dividing both sides by \[0.85\], we get
\[ \Rightarrow \dfrac{{0.30}}{{0.85}} = x\]
Thus, we get
\[ \Rightarrow x = \dfrac{{30}}{{85}} = \dfrac{6}{{17}}\]
Dividing 6 by 17 and writing the approximate value, we get
\[\begin{array}{l} \Rightarrow x \approx 0.35294\\ \Rightarrow x \approx 0.353\end{array}\]
Therefore, we get
\[P\left( B \right) = \dfrac{6}{{17}}\] or approximately \[P\left( B \right) = 0.353\]

Thus, the correct option is option (b).

Note:
We used the terms ‘union of sets’ and ‘intersection of sets’ in the solution. The intersection of two sets is the set of elements that are in both the two sets. The union of two sets is the set of all the elements that are in the two sets.
It is given that events A and B are independent. Two events are independent if the occurrence of one of the two events does not affect the probability of the occurrence of the other event.