First slide

Theorems of probability

Question

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

Moderate

A

0.54
B

0.52
C

0.38
D

0.32

Solution

Consider the following events:
A₁: A speaks truth
A₂: B speaks truth
$Then, P (A_{1}) = 60 / 100 = 3 / 5, P (A_{2}) = 70 / 100 = 7 / 10$
For the required event, either both of them should speak the
truth or both of them should tell a lie. Thus, the required probability is
$\begin{array}{l} P ((A_{1} \cap A_{2}) \cup ({\bar{A}}_{1} \cap {\bar{A}}_{2})) \\ = P (A_{1} \cap A_{2}) + P ({\bar{A}}_{1} \cap {\bar{A}}_{2}) \\ = P (A_{1}) P (A_{2}) + P ({\bar{A}}_{1}) ({\bar{A}}_{2}) \\ = \frac{3}{5} \times \frac{7}{10} + (1 - \frac{3}{5}) (1 - \frac{7}{10}) = 0 .54 \end{array}$

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