Question

A person is known to speak the truth 4 times out of 5. He throws a die and reports that it is an ace. The probability that it is actually an ace is
A.$\dfrac{1}{3}$
B.$\dfrac{2}{9}$
C.$\dfrac{4}{9}$
D.$\dfrac{5}{9}$

Answer 1

Here we will be using the concept of probability and Bayes Theorem. For this question we have two conditions, first the person speaks truth and second, he throws a die.
Whenever we have two conditions, we follow the Bayes rule

Complete step by step solution:
The rules have a very simple derivation that directly leads from the relationship between faint and conditional probabilities
Equation of Bayes rule:
$P\left( {A/B} \right) = \dfrac{{P\left( {B/A} \right)P\left( A \right)}}{{P\left( B \right)}}$
A is the event we want the probability of.
B is the new event that is related to A in the same way.
P(A/B) is called the posterior this is what we are trying to
P(B/A) is called the likelihood, this is the probability of observing the new event
A person is speaking the truth 4 times out of 5. After that he throws a die and mentions it is an ace.
Probability of speaking truth is given as: $P(A) = \dfrac{4}{5}$
Probability of getting ace from die is given as: $P(B) = \dfrac{1}{6}$
Probability of not getting ace can be calculated as:
$P(\bar A) = 1 - \dfrac{1}{6} \\\ = \dfrac{5}{6} \\\$
And, the probability of not speaking truth is calculated as:
$P(\bar B) = 1 - \dfrac{4}{5} \\\ = \dfrac{1}{5} \\\$
Probability of actually getting ace is calculated using the Bayes’ Rules:

$$P(A/B) = \dfrac{{\left( {\dfrac{4}{5} \times \dfrac{1}{6}} \right)}}{{\left( {\dfrac{4}{5} \times \dfrac{1}{6}} \right) + \left( {\dfrac{1}{5} \times \dfrac{5}{6}} \right)}} \\\ = \dfrac{{\left( {\dfrac{4}{{30}}} \right)}}{{\left( {\dfrac{4}{{30}} + \dfrac{5}{{30}}} \right)}} \\\ = \dfrac{4}{{30}} \times \dfrac{{30}}{9} \\\ = \dfrac{4}{9} \\\$$

Hence, the probability that it is actually an ace is $\dfrac{4}{9}$ .
Option C is correct.

Note: In these types of problems, Bayes’ rule should be cleared and knowledge of terms of probability is essential such as:
Event: For a random experiment, an event is any possible set of outcomes.
Outcome: An outcome of random experiment is any one of the possible results of the experiment.
Random Experiment: A random experiment is an experiment for which the set of possible outcomes is known.