Question

A speaks truth in 80% cases and B in 90% cases. The probability that they contradict each other in stating the same fact, is:

• A. \(\dfrac{17}{50}\)
• B. \(\dfrac{13}{50}\)
• C. \(\dfrac{19}{50}\)
• D. \(\dfrac{11}{50}\)

Hint:

They contradict if exactly one is truthful: \(0.8\times0.1+0.2\times0.9\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A and B contradict each other when exactly one of them speaks the truth. That can happen in two separate ways, and A and B act independently.

Step 2: Key Formula or Approach:
\(P(\text{A truth})=0.8=\dfrac45\), so \(P(\text{A lies})=\dfrac15\).
\(P(\text{B truth})=0.9=\dfrac9{10}\), so \(P(\text{B lies})=\dfrac1{10}\).

Step 3: Case 1: A tells the truth and B lies.
\[ \frac45\times\frac1{10}=\frac{4}{50} \]

Step 4: Case 2: A lies and B tells the truth.
\[ \frac15\times\frac9{10}=\frac{9}{50} \]

Step 5: Add the two cases.
The cases cannot happen together, so we add:
\[ \frac4{50}+\frac9{50}=\frac{13}{50} \]

Step 6: Check the options.
\(17/50\), \(19/50\) and \(11/50\) do not match the sum of the two cases. Only \(13/50\) is right.

Final Answer:
The probability of contradiction is \(\dfrac{13}{50}\), option 2. \[ \boxed{\dfrac{13}{50}} \]