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1. A speaks truth in 60% cases B speaks truth in 70% cases. The probability that they will way say the same thing while describing a single event, is-

A. 0.54

B. 0.56

C. 0.68

D. 0.94

Solution:

Let

E_{1}

= event that A speaks the truth
And

E_{2}

= event that B speaks the truth
Then,

\begin{aligned} P (E_{1}) = \frac{60}{100} = \frac{3}{5}, \\ P (E_{2}) = \frac{70}{100} = \frac{7}{10}, \\ P ({\bar{E}}_{1}) = (1 - \frac{3}{5}) = \frac{2}{5}, \\ P ({\bar{E}}_{2}) = (1 - \frac{7}{10}) = \frac{3}{10} \end{aligned}

P (A and B say the same thing) = P [(A speaks the truth and B speaks the truth) or (A tells a lie and B tells a lie)]

=

P [

(E_{1} \cap E_{2})

({\bar{E}}_{1} \cap {\bar{E}}_{2})]

=

(E_{1} \cap E_{2})

({\bar{E}}_{1} \cap {\bar{E}}_{2})

=

P (

E_{1}

). P (

E_{2}

) + P (

{\bar{E}}_{1}

) . P (

{\bar{E}}_{2}

)

\begin{aligned} = (\frac{3}{5} \times \frac{7}{10}) + (\frac{2}{5} \times \frac{3}{10}) \\ = \frac{27}{50} \\ = 0.54 \end{aligned}

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