Abhay speaks the truth only $$60\% .$$ Hasan rolls a die blindfolded and asks Abhay to tell him if the outcome is a ‘prime’. Abhay says, “YES”. What is the probability that the outcome is really ‘prime’ ?
A.
$$0.5$$
B.
$$0.75$$
C.
$$0.6$$
D.
none of these
Answer :
$$0.6$$
Solution :
There are two cases when Abhay will say ‘Yes’ :
$${\bf{Case}}\,\left( {\bf{i}} \right)\,{\bf{:}}$$ The number that came out is a prime and Abhay is speaking truth, probability for this case is $$P\left( P \right) \times P\left( T \right)$$
Here $$P\left( P \right) = $$ probability of getting a prime $$ = \frac{3}{6} = \frac{1}{2} = 0.5$$
$$P\left( T \right)$$ is probability that Abhay is speaking truth and $$P\left( T \right) = 0.6$$
So probability for this case is $$ = 0.5 \times 0.6 = 0.3$$
$${\bf{Case}}\,\left( {{\bf{ii}}} \right)\,{\bf{:}}$$ The number that came out is not a prime and Abhay is not speaking truth, probability for this case is $$P\left( {P'} \right) \times P\left( {T'} \right) = 0.5 \times 0.4 = 0.2$$
So total probability for the given case is $$ = 0.3 + 0.2 = 0.5$$
New sample space is $$0.5$$ and we have to find the probability of case $$\left( {\text{i}} \right)$$ which is $$ = \frac{{0.3}}{{0.5}} = 0.6$$
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.
$$\frac{{P\left( {\overline A } \right)}}{{P\left( {\overline B } \right)}}$$