A speaks the truth 5 out of 7 times and B speaks truth 8 out of 9 times. What is the probability that they contradict each other in stating the same fact?
Understanding the Problem: Probability of Contradiction
This question asks for the probability that two individuals, A and B, contradict each other when stating the same fact. We are given the individual probabilities of each person speaking the truth.
Contradiction occurs when one person speaks the truth and the other person lies about the same fact.
Calculating Individual Probabilities
First, let's determine the probabilities of A and B speaking the truth and lying.
- Probability that A speaks the truth, denoted as P(A truth):
\(\text{P(A truth)} = \frac{5}{7}\)
- Probability that A lies, denoted as P(A lie):
\(\text{P(A lie)} = 1 - \text{P(A truth)} = 1 - \frac{5}{7} = \frac{7-5}{7} = \frac{2}{7}\)
- Probability that B speaks the truth, denoted as P(B truth):
\(\text{P(B truth)} = \frac{8}{9}\)
- Probability that B lies, denoted as P(B lie):
\(\text{P(B lie)} = 1 - \text{P(B truth)} = 1 - \frac{8}{9} = \frac{9-8}{9} = \frac{1}{9}\)
Conditions for Contradiction
Two people contradict each other on a fact if:
- Case 1: A speaks the truth AND B lies.
- Case 2: A lies AND B speaks the truth.
Since A's statement and B's statement are independent events (one person's truth-telling doesn't affect the other's), we can multiply their individual probabilities for each case.
Calculating Probability of Contradiction
Let's calculate the probability for each case and then sum them up.
- Probability of Case 1 (A truth and B lie):
\(\text{P(A truth and B lie)} = \text{P(A truth)} \times \text{P(B lie)} = \frac{5}{7} \times \frac{1}{9} = \frac{5 \times 1}{7 \times 9} = \frac{5}{63}\)
- Probability of Case 2 (A lie and B truth):
\(\text{P(A lie and B truth)} = \text{P(A lie)} \times \text{P(B truth)} = \frac{2}{7} \times \frac{8}{9} = \frac{2 \times 8}{7 \times 9} = \frac{16}{63}\)
The total probability that they contradict each other is the sum of the probabilities of these two mutually exclusive cases:
\(\text{P(Contradiction)} = \text{P(A truth and B lie)} + \text{P(A lie and B truth)}\)
\(\text{P(Contradiction)} = \frac{5}{63} + \frac{16}{63} = \frac{5 + 16}{63} = \frac{21}{63}\)
Simplifying the fraction:
\(\frac{21}{63} = \frac{21 \div 21}{63 \div 21} = \frac{1}{3}\)
Summary of Contradiction Probability
The probability that A and B contradict each other in stating the same fact is \(\frac{1}{3}\).
Revision Table: Probability Concepts
| Concept | Description | Formula Example |
|---|---|---|
| Probability of an Event | Likelihood of an event occurring. | P(E) = (Number of favorable outcomes) / (Total number of outcomes) |
| Complementary Events | An event happening vs. not happening. | P(E') = 1 - P(E) |
| Independent Events | Outcome of one event does not affect the other. | P(A and B) = P(A) × P(B) |
| Mutually Exclusive Events | Events that cannot happen at the same time. | P(A or B) = P(A) + P(B) |
Additional Information: Independent Events
In this problem, we assume that A's tendency to speak the truth or lie is completely independent of B's tendency. This means that whether A speaks the truth on a given fact does not influence whether B speaks the truth on the same fact. Because of this independence, we were able to multiply their individual probabilities (e.g., P(A truth) * P(B lie)) to find the probability of both events happening together (a joint event).
Real-world scenarios might involve dependence (e.g., if they are colluding), but for this type of probability question, independence is the standard assumption unless otherwise stated.
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