Explanation
Let us define the events for both individuals speaking the truth or lying.
Step 1: Determine Individual Probabilities
For Person A:
Probability that A speaks the truth, P(A)=40%=10040=52
Probability that A lies (tells a falsehood), P(A′)=1−P(A)=1−52=53
For Person B:
Probability that B speaks the truth, P(B)=50%=10050=21
Probability that B lies (tells a falsehood), P(B′)=1−P(B)=1−21=21
Step 2: Understand the Condition for Contradiction
Two people will contradict each other if one of them speaks the truth and the other one lies. Since their speaking habits are independent events, this can happen in two mutually exclusive cases:
Case 1: A speaks the truth AND B lies (A∩B′)
Case 2: A lies AND B speaks the truth (A′∩B)
Step 3: Formulate and Calculate the Total Probability
The total probability of contradiction is given by:
P(Contradiction)=P(A∩B′)+P(A′∩B)
Since the events are independent, we can multiply their individual probabilities:
P(Contradiction)=[P(A)×P(B′)]+[P(A′)×P(B)]
Substitute the values calculated in Step 1 into this equation:
P(Contradiction)=(52×21)+(53×21)
P(Contradiction)=102+103
P(Contradiction)=105=21
