(i) P speaks truth in 70% of the cases and Q in 80% of the cases. In what persentage of cases are they likely to agree in stating the same fact ?
(ii) A speaks truth in 75% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact ?

Answer

To solve the given problems step by step, we will break down each part of the question.

### Part (i)
**Question:** P speaks truth in 70% of the cases and Q in 80% of the cases. In what percentage of cases are they likely to agree in stating the same fact?

1. **Identify the probabilities:**
- Probability that P speaks the truth, \( P(T_P) = 0.7 \)
- Probability that Q speaks the truth, \( P(T_Q) = 0.8 \)

2. **Calculate the probability that both P and Q speak the truth:**
\[ P(T_P \cap T_Q) = P(T_P) \times P(T_Q) = 0.7 \times 0.8 = 0.56 \]

3. **Calculate the probabilities that both P and Q lie:**
- Probability that P lies, \( P(L_P) = 1 - P(T_P) = 1 - 0.7 = 0.3 \)
- Probability that Q lies, \( P(L_Q) = 1 - P(T_Q) = 1 - 0.8 = 0.2 \)
\[ P(L_P \cap L_Q) = P(L_P) \times P(L_Q) = 0.3 \times 0.2 = 0.06 \]

4. **Total probability that P and Q agree:**
\[ P(\text{Agree}) = P(T_P \cap T_Q) + P(L_P \cap L_Q) = 0.56 + 0.06 = 0.62 \]

5. **Convert to percentage:**
\[ P(\text{Agree}) = 0.62 \times 100 = 62\% \]

### Part (ii)
**Question:** A speaks truth in 75% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact?

1. **Identify the probabilities:**
- Probability that A speaks the truth, \( P(T_A) = 0.75 \)
- Probability that B speaks the truth, \( P(T_B) = 0.90 \)

2. **Calculate the probabilities that A lies and B lies:**
- Probability that A lies, \( P(L_A) = 1 - P(T_A) = 1 - 0.75 = 0.25 \)
- Probability that B lies, \( P(L_B) = 1 - P(T_B) = 1 - 0.90 = 0.10 \)

3. **Calculate the probabilities of contradiction:**
- Case 1: A tells the truth and B lies:
\[ P(T_A \cap L_B) = P(T_A) \times P(L_B) = 0.75 \times 0.10 = 0.075 \]
- Case 2: A lies and B tells the truth:
\[ P(L_A \cap T_B) = P(L_A) \times P(T_B) = 0.25 \times 0.90 = 0.225 \]

4. **Total probability of contradiction:**
\[ P(\text{Contradict}) = P(T_A \cap L_B) + P(L_A \cap T_B) = 0.075 + 0.225 = 0.30 \]

5. **Convert to percentage:**
\[ P(\text{Contradict}) = 0.30 \times 100 = 30\% \]

### Final Answers:
(i) They are likely to agree in stating the same fact in **62%** of cases.
(ii) They are likely to contradict each other in stating the same fact in **30%** of cases.