A bag contains $$n + 1$$ coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is $$\frac{7}{{12}},$$ then the value of $$n$$ is :
A.
$$3$$
B.
$$4$$
C.
$$5$$
D.
none of these
Answer :
$$5$$
Solution :
Let $${E_1}$$ denote the event "a coin with head on both sides is selected" and $${E_2}$$ denotes the event "a fair coin is selected".
Let $$A$$ be the event "he toss, results in heads".
$$\eqalign{
& \therefore \,P\left( {{E_1}} \right) = \frac{1}{{n + 1}},\,P\left( {{E_2}} \right) = \frac{n}{{n + 1}}{\text{ and}} \cr
& P\left( {\frac{A}{{{E_1}}}} \right) = 1,\,P\left( {\frac{A}{{{E_2}}}} \right) = \frac{1}{2} \cr
& \therefore \,P\left( A \right) = P\left( {{E_1}} \right)P\left( {\frac{A}{{{E_1}}}} \right) + P\left( {{E_2}} \right)P\left( {\frac{A}{{{E_2}}}} \right) \cr
& \Rightarrow \frac{7}{{12}} = \frac{1}{{n + 1}} \times 1 + \frac{n}{{n + 1}} \times \frac{1}{2} \cr
& \Rightarrow n = 5 \cr} $$
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.
$$\frac{{P\left( {\overline A } \right)}}{{P\left( {\overline B } \right)}}$$