Question

The probability of simultaneous occurrence of 2 events A & B is \[p\]. If the probability that exactly one of A, B occurs is \[q\], then which of the following alternatives is INCORRECT?
A.\[P\left( {\overline A } \right) + P\left( {\overline B } \right) = 2 + 2q - p\]
B.\[P\left( {\overline A } \right) + P\left( {\overline B } \right) = 2 - 2p - q\]
C.\[P\left( {\left( {A \cap B} \right)/\left( {A \cup B} \right)} \right) = \frac{p}{{p + q}}\]
D.\[P\left( {\overline A \cap \overline B } \right) = 21 - p - q\]

Answer 1

Hint: Here we have to use the basic formulas of the probability which is given in the options and then put the given values of the probability in those formulas to check the conditions or the values of all the options whether it is correct or incorrect.

Complete step-by-step answer:
It is given that the probability of simultaneous occurrence of 2 events A & B is \[p\] i.e. \[P\left( {A \cap B} \right) = p\] and probability that exactly one of A, B occurs is \[q\]i.e. \[P\left( A \right) + P\left( B \right) - 2P\left( {A \cap B} \right) = q\]
Firstly we will find the value for the formula of the probability given in the option A i.e. \[P\left( {\overline A } \right) + P\left( {\overline B } \right)\]
As we know that
\[P\left( A \right) + P\left( B \right) - 2P\left( {A \cap B} \right) = q\]
So we will put the value of \[P\left( {A \cap B} \right) = p\]in this equation, we get
\[ \Rightarrow P\left( A \right) + P\left( B \right) - 2p = q\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) = 2p + q\]
Now we will write these probabilities in the form of their complementary function which is equal to the difference of that probability from 1. So, we get
\[ \Rightarrow 1 - P\left( {\overline A } \right) + 1 - P\left( {\overline B } \right) = 2p + q\]
Now we will simplify this equation
\[ \Rightarrow P\left( {\overline A } \right) + P\left( {\overline B } \right) = 2 - 2p - q\]
Hence, from this we can say that the value of \[P\left( {\overline A } \right) + P\left( {\overline B } \right)\] is equal to \[2 - 2p - q\]. So, from this we can say that option A is incorrect and option B is correct.
So, option A is incorrect.

Note: Here, we should keep in mind at least the basic formula of the probability for solving the probability type of questions.
\[ \Rightarrow P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\]
We should know that the probability of occurrence is always less than or equal to one. If the probability of the event is one then that event is known as the true event. We should also note that the sum of the probability of occurrence of an event and sum of the probability of not occurrence of an event is always equal to one. We used the same concept to find our required probability. Probability is the subject which helps us a lot in the prediction of the events.