The probability that a particular day in the month of July is a rainy day is $$\frac{3}{4}$$. Two person whose credibility are $$\frac{4}{5}$$ and $$\frac{2}{3}$$, respectively, claim that $${15^{th}}$$ July was a rainy day. The probability that it was really a rainy day is :
A.
$$\frac{{12}}{{13}}$$
B.
$$\frac{{11}}{{12}}$$
C.
$$\frac{{24}}{{25}}$$
D.
$$\frac{{29}}{{30}}$$
Answer :
$$\frac{{24}}{{25}}$$
Solution :
Let events
$$A :$$ Event that first man speaks truth
$$B :$$ Event that second man speaks truth
$$R :$$ Day is rainy
$$\therefore \,P\left( A \right) = \frac{4}{5},\,P\left( B \right) = \frac{2}{3},\,P\left( R \right) = \frac{3}{4}$$
$$\therefore $$ Required probability
$$\eqalign{
& = \frac{{P\left( {A \cap B} \right).P\left( R \right)}}{{P\left( {A \cap B} \right).P\left( R \right) + P\left( {A' \cap B'} \right).P\left( {R'} \right)}} \cr
& = \frac{{\frac{4}{5} \times \frac{2}{3} \times \frac{3}{4}}}{{\frac{4}{5} \times \frac{2}{3} \times \frac{3}{4} + \frac{1}{5} \times \frac{1}{3} \times \frac{1}{4}}} \cr
& = \frac{{24}}{{25}} \cr} $$
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.
$$\frac{{P\left( {\overline A } \right)}}{{P\left( {\overline B } \right)}}$$