A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and IV. The probabilities of the student passing in tests I, II, III are $$p,\,q$$ and $$\frac{1}{2}$$ respectively. The probability that the student is successful is $$\frac{1}{2}$$ then the relation between $$p$$ and $$q$$ is given by :
A.
$$pq + p = 1$$
B.
$${p^2} + q = 1$$
C.
$$pq - 1 = p$$
D.
none of these
Answer :
$$pq + p = 1$$
Solution :
Let $$A,\,B$$ and $$C$$ be the events that the student is successful in tests I, II and III respectively.
Then $$P$$ (The student is successful)
$$\eqalign{
& = P\left( A \right)P\left( B \right)\left\{ {1 - P\left( C \right)} \right\} + P\left( A \right)\left\{ {1 - P\left( B \right)} \right\}P\left( C \right) + P\left( A \right)P\left( B \right)P\left( C \right) \cr
& = p.q\left( {1 - \frac{1}{2}} \right) + p\left( {1 - q} \right)\frac{1}{2} + p.q\frac{1}{2} \cr
& = \frac{1}{2}pq + \frac{1}{2}p\left( {1 - q} \right) + \frac{1}{2}pq \cr
& = \frac{1}{2}\left( {pq + p - pq + pq} \right) \cr
& = \frac{1}{2}\left( {pq + p} \right) \cr
& \therefore \,\frac{1}{2} = \frac{1}{2}\left( {pq + p} \right) \Rightarrow 1 = pq + p \cr} $$
Releted MCQ Question on Statistics and Probability >> Probability
Releted Question 1
Two fair dice are tossed. Let $$x$$ be the event that the first die shows an even number and $$y$$ be the event that the second die shows an odd number. The two events $$x$$ and $$y$$ are:
Two events $$A$$ and $$B$$ have probabilities 0.25 and 0.50 respectively. The probability that both $$A$$ and $$B$$ occur simultaneously is 0.14. Then the probability that neither $$A$$ nor $$B$$ occurs is
The probability that an event $$A$$ happens in one trial of an experiment is 0.4. Three independent trials of the experiment are performed. The probability that the event $$A$$ happens at least once is
If $$A$$ and $$B$$ are two events such that $$P(A) > 0,$$ and $$P\left( B \right) \ne 1,$$ then $$P\left( {\frac{{\overline A }}{{\overline B }}} \right)$$ is equal to
(Here $$\overline A$$ and $$\overline B$$ are complements of $$A$$ and $$B$$ respectively).
A.
$$1 - P\left( {\frac{A}{B}} \right)$$
B.
$$1 - P\left( {\frac{{\overline A }}{B}} \right)$$
C.
$$\frac{{1 - P\left( {A \cup B} \right)}}{{P\left( {\overline B } \right)}}$$
D.
$$\frac{{P\left( {\overline A } \right)}}{{P\left( {\overline B } \right)}}$$