Question

Two person, whose probabilities of speaking the truth are $\dfrac{2}{3}$ and $\dfrac{5}{6}$ respectively, assert that a specified ticket has been drawn out of a bag containing $15$ tickets. What is the probability of the truth of the assertion?

Answer 1

When we try to find the probability of any event then we need to find the all possible event and the sample of event which can occur in the probability of the given event or we can say the number of favourable events.

Complete step by step Solution:
We know the the formula for finding the probability is the
$P(A) =$ Number of Favourable Event / Total Number of Favourable Event
And the formula of the sum of Possible Event + Not Possible Event $= 1$
Now given that probabilities of speaking the truth are $\dfrac{2}{3}$ and $\dfrac{5}{6}$
Now we will find the value of the false speaking probability then we will get
First person speaking false then the probability is the
$= 1 - \dfrac{2}{3} = \dfrac{1}{3}$
Second person speaking false then the probability is the
$= 1 - \dfrac{5}{6} = \dfrac{1}{6}$
Then now we will find the probability of the speaking of truth is the
= Both person speaking truth /(one truth,second truth)+(one truth,second false)+(one false,second truth)+(both first ,second truth)
$= \dfrac{{\dfrac{2}{3} \times \dfrac{5}{6}}}{{\dfrac{2}{3} \times \dfrac{5}{6} + \dfrac{1}{3} \times \dfrac{2}{6} + \dfrac{2}{3} \times \dfrac{1}{6} + \dfrac{1}{3} \times \dfrac{1}{6}}}$
Now the value of the speaking the truth is the
$= \dfrac{5}{7}$

Therefore the probability of our question is the $\dfrac{5}{7}$

Note:
We know the all formula for finding the probability but when we need to find the probability of two events which occur simultaneously then we will use the above way to solve those questions.