A dice is thrown $$2n + 1$$ times, $$n\, \in \,N.$$ The probability that faces with even numbers show odd number of times is :
A.
$$\frac{{2n + 1}}{{4n + 3}}$$
B.
less than $$\frac{1}{2}$$
C.
greater than $$\frac{1}{2}$$
D.
none of these
Answer :
none of these
Solution :
The probability of showing an even number in a throw $$ = \frac{3}{6} = \frac{1}{2}$$
$$\therefore $$ the required probability
$$\eqalign{
& = {}^{2n + 1}{C_1}.\frac{1}{2}.{\left( {\frac{1}{2}} \right)^{2n}} + {}^{2n + 1}{C_3}.{\left( {\frac{1}{2}} \right)^3}.{\left( {\frac{1}{2}} \right)^{2n - 2}} + ...... + {}^{2n + 1}{C_{2n + 1}}.{\left( {\frac{1}{2}} \right)^{2n + 1}} \cr
& = \left( {{}^{2n + 1}{C_1} + {}^{2n + 1}{C_3} + ...... + {}^{2n + 1}{C_{2n + 1}}} \right).{\left( {\frac{1}{2}} \right)^{2n + 1}} \cr
& = {2^{2n}}.{\left( {\frac{1}{2}} \right)^{2n + 1}} \cr
& = \frac{1}{2} \cr} $$
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.
$$\frac{{P\left( {\overline A } \right)}}{{P\left( {\overline B } \right)}}$$