Question

‘A’ speaks truth in 60% of the cases and ‘B’ in 90% of the cases. The percentage of cases they are likely to contradict each other in stating the same fact is

• A. 0.42
• B. 0.24
• C. 0.5
• D. 0.32

Answer 1

The correct option is A 0.42

We want to find the probability of A and B contradicting each other when they make a statement. Think about the possible ways this can happen. What are the possibilities when they make a statement?
Before we start writing them down, lets first give names for some events.
Let E be the event A speaks truth.
Let F be the event B speaks truth.
When both of them make a statement, the possible outcomes are EF, E’F, EF’, E’F’
Here, E’ and F’ correspond to the events A tells a lie and B speaks a lie.
Out of these, the events E’F and EF’ corresponds to the cases ‘A tells a lie and B tells truth’ and ‘B tells a lie and A tells truth’. In these cases A and B will contradict each other. When they both tells lie or truth, they will not contradict with each other.
Note that A telling a lie is not dependent on if B tells lie or truth. So we can say they are independent events.
$P\left(E\right)=\frac{60}{100}=\frac{3}{5}$
$P\left(F\right)=\frac{90}{100}=\frac{9}{10}$
Probability of A and B contradicting with each other = P(E’ F) + P(EF’)
Since E and F are independent events, we can write the probability as
Probability of A and B contradicting with each other = P(E’ F) + P(EF’)
= P(E’) P(F) + P(E) P(F’)
$=(1-\frac{3}{5})\frac{9}{10}+\frac{3}{5}(1-\frac{9}{10})$
$=\frac{21}{50}$
$=0.42$