Atal speaks the truth in 70% of the cases and George speaks the truth in 60% cases. In what percentage of cases are they likely to contradict each other in stating the same fact?
46%
Understanding the Problem: Contradiction Probability
This problem asks us to find the likelihood that two individuals, Atal and George, will contradict each other when stating the same fact. We are given the probabilities of each person speaking the truth.
Probabilities Involved
First, let's list the probabilities provided:
- Probability that Atal speaks the truth: Let this be $P(A_{truth}) = 70\% = 0.70$.
- Probability that Atal lies: This is the complement of speaking the truth, so $P(A_{lie}) = 1 - P(A_{truth}) = 1 - 0.70 = 0.30$ (or 30%).
- Probability that George speaks the truth: Let this be $P(G_{truth}) = 60\% = 0.60$.
- Probability that George lies: This is the complement of speaking the truth, so $P(G_{lie}) = 1 - P(G_{truth}) = 1 - 0.60 = 0.40$ (or 40%).
Scenarios for Contradiction
Atal and George contradict each other in two possible scenarios when they state the same fact:
- Scenario 1: Atal tells the truth, and George lies.
- Scenario 2: Atal lies, and George tells the truth.
Calculating the Probability of Each Scenario
We assume that Atal's and George's statements are independent events. Therefore, we can multiply their individual probabilities.
Scenario 1 Calculation
The probability that Atal tells the truth AND George lies is:
$$ P(A_{truth} \text{ and } G_{lie}) = P(A_{truth}) \times P(G_{lie}) $$
$$ P(A_{truth} \text{ and } G_{lie}) = 0.70 \times 0.40 = 0.28 $$
Scenario 2 Calculation
The probability that Atal lies AND George tells the truth is:
$$ P(A_{lie} \text{ and } G_{truth}) = P(A_{lie}) \times P(G_{truth}) $$
$$ P(A_{lie} \text{ and } G_{truth}) = 0.30 \times 0.60 = 0.18 $$
Total Probability of Contradiction
Since these two scenarios are mutually exclusive (they cannot happen at the same time), the total probability of them contradicting each other is the sum of the probabilities of the two scenarios.
$$ P(\text{Contradiction}) = P(A_{truth} \text{ and } G_{lie}) + P(A_{lie} \text{ and } G_{truth}) $$
$$ P(\text{Contradiction}) = 0.28 + 0.18 = 0.46 $$
Expressing the Result as a Percentage
To express this probability as a percentage, we multiply by 100:
$$ \text{Percentage Contradiction} = 0.46 \times 100\% = 46\% $$
Conclusion
The percentage of cases where Atal and George are likely to contradict each other is 46%.
Important Questions from Multiplication Theorem of Events
The probability that A speaks truth is 4 / 5 while this probability for B is 3 / 4. The probability that they contradict each other when asked to speak on a fact is
For any two events A and B, the probability that at least one of them occur is 0.6. If A and B occur simultaneously with a probability 0.3, then P(A') + P(B') is
A student appears for tests I, II and III. The student is considered successful if the passes in tests I, II or I, III or all the three. The probabilities of the student passing in test I, II and III are m, n and 1/2 respectively. If the probability of the student to be successful is 1/2, then which one of the following is correct?
If \(\rm P(A\cup B)=\dfrac{5}{6}, P(A\cap B)=\dfrac{1}{3}\:and\:P(\bar A)=\dfrac{1}{2}\) , then which of the following is/are correct?
1. A and B are independent events.
2. A and B are mutually exclusive events.
Select the correct answer using the code given below.
In a lottery of 10 tickets numbered 1 to 10, two tickets are drawn simultaneously. What is the probability that both the tickets drawn have prime numbers?
