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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
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B
0.24
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C
0.5
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D
0.32
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Solution

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(E)=60100=35
P(F)=90100=910
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’)
=(135)910+35(1910)
=2150
=0.42

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