Find the probability that they speak the same fact if A speaks truth in 60% cases and B speaks truth in 75% cases?
Probability of Speaking the Same Fact: A Detailed Explanation
This problem asks us to find the probability that two individuals, A and B, speak the same fact, given their individual truth-telling percentages. Understanding the probabilities of them speaking the truth and lying is crucial to solving this problem.
Understanding the Given Probabilities
We are provided with the following information:
- Individual A speaks the truth in 60% cases.
- Individual B speaks the truth in 75% cases.
Let's define the probabilities for each individual:
- Probability that A speaks the truth, denoted as \(P(T_A)\): \(P(T_A) = 60\% = \frac{60}{100} = 0.60\)
- Probability that A lies (does not speak the truth), denoted as \(P(L_A)\): Since A either speaks the truth or lies, \(P(L_A) = 1 - P(T_A)\). \(P(L_A) = 1 - 0.60 = 0.40\)
- Probability that B speaks the truth, denoted as \(P(T_B)\): \(P(T_B) = 75\% = \frac{75}{100} = 0.75\)
- Probability that B lies (does not speak the truth), denoted as \(P(L_B)\): Since B either speaks the truth or lies, \(P(L_B) = 1 - P(T_B)\). \(P(L_B) = 1 - 0.75 = 0.25\)
Conditions for Speaking the Same Fact
For A and B to speak the same fact, there are two possible scenarios:
- Both speak the truth: This means A speaks the truth AND B speaks the truth.
- Both lie: This means A lies AND B lies.
Since the actions of A and B are independent (one person's truth-telling doesn't affect the other's), we can multiply their individual probabilities for combined events.
Calculating Probabilities for Each Scenario
Probability of Both Speaking Truth
The probability that both A and B speak the truth is the product of their individual probabilities of speaking the truth:
\[P(\text{A speaks truth AND B speaks truth}) = P(T_A) \times P(T_B)\] \[P(\text{Both Truth}) = 0.60 \times 0.75\] \[P(\text{Both Truth}) = 0.45\]Probability of Both Lying
The probability that both A and B lie is the product of their individual probabilities of lying:
\[P(\text{A lies AND B lies}) = P(L_A) \times P(L_B)\] \[P(\text{Both Lie}) = 0.40 \times 0.25\] \[P(\text{Both Lie}) = 0.10\]Finding the Overall Probability of Speaking the Same Fact
The total probability that they speak the same fact is the sum of the probabilities of these two mutually exclusive scenarios (both speak truth, or both lie):
\[P(\text{Speak Same Fact}) = P(\text{Both Truth}) + P(\text{Both Lie})\] \[P(\text{Speak Same Fact}) = 0.45 + 0.10\] \[P(\text{Speak Same Fact}) = 0.55\]To express this as a percentage, we multiply by 100:
\[P(\text{Speak Same Fact}) = 0.55 \times 100\% = 55\%\]Summary of Calculation
| Event | Probability | Calculation |
|---|---|---|
| A speaks truth | \(P(T_A) = 0.60\) | Given |
| A lies | \(P(L_A) = 0.40\) | \(1 - 0.60\) |
| B speaks truth | \(P(T_B) = 0.75\) | Given |
| B lies | \(P(L_B) = 0.25\) | \(1 - 0.75\) |
| Both speak truth | \(P(\text{Both Truth}) = 0.45\) | \(P(T_A) \times P(T_B) = 0.60 \times 0.75\) |
| Both lie | \(P(\text{Both Lie}) = 0.10\) | \(P(L_A) \times P(L_B) = 0.40 \times 0.25\) |
| Speak the same fact | \(P(\text{Same Fact}) = 0.55\) | \(P(\text{Both Truth}) + P(\text{Both Lie}) = 0.45 + 0.10\) |
Therefore, the probability that they speak the same fact is 55%.
Important Questions from Conditional Probability
Two events A and B are such that P(not B) = 0.8, P(A ∪ B) = 0.5 and P(A|B) = 0.4. Then P(A) is equal to
For two events, A and B, it is given that \({\rm{P}}\left( {\rm{A}} \right) = \frac{3}{5},{\rm{\;P}}\left( {\rm{B}} \right) = \frac{3}{{10}}\) and \({\rm{P}}\left( {{\rm{A|B}}} \right) = \frac{2}{3}\) . If A̅ and B̅ are the complementary events of A and B, then what is P(A̅ | B̅) equal to?
For two mutually exclusive events A and B, P(A) = 0.2 and P (A̅ ∩ B) = 0.3. What is P (A|(A ∪ B)) equal to?
If an event B has occurred and has P(B) = 1, the conditional probability P(A|B) is equal to:
If P(A) = 0.7, P(B) = 0.5 and P(B/A) = 0.3, find (i) P(A/B) (ii) P(A ∪ B)?
