The probabilities that persons A and B will tell the truth about an event are 2/3 and 4/5 respectively. Assuming their statements are independent, what is the probability that they will agree with each other?

Introduction / Context:
This question examines agreement between two people who may or may not tell the truth. Each person has a different probability of telling the truth, and their statements about the same event are assumed to be independent. They are said to agree when both tell the truth or both tell lies. The problem therefore involves calculating the probability of these two joint cases and adding them.

Given Data / Assumptions:

• Probability that A tells the truth = 2/3.
• Probability that A tells a lie = 1 - 2/3 = 1/3.
• Probability that B tells the truth = 4/5.
• Probability that B tells a lie = 1 - 4/5 = 1/5.
• The behaviour of A and B is independent.
• They agree if both statements are true or both are false.

Concept / Approach:
Let T denote telling the truth and L denote telling a lie. Agreement occurs in exactly two cases: A is T and B is T, or A is L and B is L. Since these cases are mutually exclusive, the total probability of agreement is the sum of their individual probabilities. Because A and B act independently, we can find each joint probability by multiplying the corresponding individual probabilities.

Step-by-Step Solution:
P(A tells truth) = 2/3, P(A tells lie) = 1/3. P(B tells truth) = 4/5, P(B tells lie) = 1/5. Case 1: Both tell truth. P(both truth) = (2/3) * (4/5) = 8 / 15. Case 2: Both tell lies. P(both lies) = (1/3) * (1/5) = 1 / 15. Total probability of agreement = 8 / 15 + 1 / 15 = 9 / 15. Simplify 9 / 15 by dividing numerator and denominator by 3 to get 3 / 5.

Verification / Alternative check:
We can use the complement idea by first finding the probability that they contradict each other. Contradiction occurs if A tells truth and B lies, or A lies and B tells truth. P(contradiction) = (2/3)*(1/5) + (1/3)*(4/5) = 2/15 + 4/15 = 6/15 = 2/5. Since agreement and contradiction cover all possibilities, P(agreement) = 1 minus 2/5 = 3/5, confirming the earlier result.

Why Other Options Are Wrong:
1/3 and 2/5 do not match the combined probability of the two agreement cases and instead relate to partial parts of the outcome space. 3/4 suggests 75 percent agreement, which is too high given that both individuals have a significant chance of lying.

Common Pitfalls:
Learners sometimes assume agreement means both tell the truth only, forgetting that both can also lie and still give matching statements. Another common error is to add the individual truth probabilities directly rather than multiplying to get joint probabilities. Drawing a small table of outcomes TT, TL, LT and LL helps to visualize when agreement happens and ensures the correct cases are counted.

Final Answer:
The probability that A and B agree with each other is 3/5.