A speaks truth in 70 percent of cases and B in 90 percent cases. In what percentage of cases are they likely to contradict each other in stating the same fact?

To find the percentage of cases in which A and B are likely to contradict each other when stating the same fact, we need to consider the cases where one tells the truth and the other does not.

### Step 1: Determine the Probabilities
- Probability that A speaks the truth: \( P(A_{\text{truth}}) = 0.7 \)
- Probability that A lies: \( P(A_{\text{lie}}) = 1 - 0.7 = 0.3 \)
- Probability that B speaks the truth: \( P(B_{\text{truth}}) = 0.9 \)
- Probability that B lies: \( P(B_{\text{lie}}) = 1 - 0.9 = 0.1 \)

### Step 2: Cases of Contradiction
A and B will contradict each other in two scenarios:
1. A tells the truth and B lies.
2. A lies and B tells the truth.

#### Case 1: A tells the truth and B lies
- Probability: \( P(A_{\text{truth}}) \times P(B_{\text{lie}}) = 0.7 \times 0.1 = 0.07 \)

#### Case 2: A lies and B tells the truth
- Probability: \( P(A_{\text{lie}}) \times P(B_{\text{truth}}) = 0.3 \times 0.9 = 0.27 \)

### Step 3: Add the Probabilities
To find the total probability that they contradict each other:

\[
P(\text{contradiction}) = P(A_{\text{truth}} \text{ and } B_{\text{lie}}) + P(A_{\text{lie}} \text{ and } B_{\text{truth}}) = 0.07 + 0.27 = 0.34
\]

### Final Answer:
They are likely to contradict each other in 34% of the cases.

---

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What is the probability that both A and B state the truth simultaneously?
2. How does the result change if A and B have different probabilities of speaking the truth?
3. What is the probability that both A and B lie simultaneously?
4. How would this problem change if there were three people instead of two?
5. What if A and B had complementary probabilities of speaking the truth and lying?

**Tip:** When calculating combined probabilities, it's often easier to first identify the individual scenarios where the desired outcome occurs, then sum the probabilities of these scenarios.

Explore the probability of contradiction between A and B when stating the same fact. Understand how to calculate probabilities when individuals have different truth-telling probabilities. Suitable for advanced high school students studying probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation