If A speaks truth in 60% of the cases while B speaks truth in 75% of the case. Then, find the probability that they contradict in stating the same fact (rounded up to two decimal places).Correct answer is between '0.42,0.47'. Can you explain this answer?

Question

Answer

Let the probability that A and B speak truth be P(A) and P(B) resp.
P(A) =3/5 and P(B)=3/4

Case 1 : A speaks truth and B does not
The required probability = P(A) * (1- P(B))
= 3/5 * (1-3/4)
= 3/20
Case 2 : B speaks truth and A does not
The required probability = (1- P(A)) * P(B)
=(1- 3/5) * 3/4
=6/20
Hence , the percentage of cases in which they are likely to contradict in stating the same fact = (3/20 + 6/20) * 100%
= 9/20 * 100% = 45%
here the answer 0.45 is in between 0.42 and 0.47

Answer

Understanding the Problem:
We are given that person A speaks the truth in 60% of the cases and person B speaks the truth in 75% of the cases. We need to find the probability that they contradict each other when stating the same fact.

Approach:
To solve this problem, we can consider all the possible cases where A and B can contradict each other. There are two possibilities: either A speaks the truth and B speaks a lie, or A speaks a lie and B speaks the truth.

Calculating the Probability:
We can calculate the probability of A speaking the truth and B speaking a lie using the following formula:
P(A speaks truth and B speaks lie) = P(A speaks truth) * P(B speaks lie)

Given that A speaks the truth in 60% of the cases and B speaks a lie in 25% of the cases, we can substitute these values into the formula:
P(A speaks truth and B speaks lie) = 0.60 * 0.25 = 0.15

Similarly, we can calculate the probability of A speaking a lie and B speaking the truth using the formula:
P(A speaks lie and B speaks truth) = P(A speaks lie) * P(B speaks truth)

Given that A speaks a lie in 40% of the cases and B speaks the truth in 75% of the cases, we can substitute these values into the formula:
P(A speaks lie and B speaks truth) = 0.40 * 0.75 = 0.30

Calculating the Final Probability:
To calculate the probability that they contradict each other, we need to add the probabilities of the two cases:
P(contradiction) = P(A speaks truth and B speaks lie) + P(A speaks lie and B speaks truth)

Substituting the calculated values, we get:
P(contradiction) = 0.15 + 0.30 = 0.45

As the answer needs to be rounded up to two decimal places, we round 0.45 to 0.47.

Therefore, the probability that A and B contradict each other when stating the same fact is between 0.42 and 0.47.

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