Question

A speaks truth in \(60 \%\) cases and \(B\) speaks truth in \(80 \%\) cases. The probability that they will say the same thing while describing a single event is : (a) \(0.36\) (b) \(0.56\) (c) \(0.48\) (d) \(0.20\)

Short answer

Answer: The probability that A and B will say the same thing while describing a single event is 0.56.

Step-by-step solution

Determine the probability of A and B both telling the truth.

To find the probability of A and B both telling the truth, we simply multiply their individual probabilities of telling the truth. In this case, A has a 60% chance of telling the truth, and B has an 80% chance of telling the truth: Probability of both telling the truth = \(0.60 \times 0.80 = 0.48\)

Determine the probability of A and B both lying.

Similarly, we need to find the probability of both A and B lying. To do this, we'll consider the probabilities of them not telling the truth. Since A tells the truth 60% of the time, they lie in 40% of the cases. Likewise, B tells the truth 80% of the time, so they lie 20% of the time: Probability of both lying = \(0.40 \times 0.20 = 0.08\)

Calculate the probability of A and B saying the same thing.

Now that we have the probabilities of A and B both telling the truth and both lying, we can add them together to find the probability of them saying the same thing: Probability of A and B saying the same thing = Probability of both telling the truth + Probability of both lying = \(0.48 + 0.08 = 0.56\) The answer is: (b) \(0.56\)

Key concepts

Joint Probability

Joint probability is a fundamental concept in the realm of probability. It refers to the chance of two or more events happening simultaneously. In simpler terms, it answers the question: what's the likelihood of both events occurring at the same time?
For example, if we want to know the probability that two people, say A and B, both tell the truth, we are essentially looking for the joint probability of A and B speaking honestly.
This typically involves multiplying the probabilities of each individual event. So, for A telling the truth and B also telling the truth, we multiply their probabilities:

• A's probability of truth: 0.60
• B's probability of truth: 0.80

Joint probability is significant in scenarios where the occurrence of one event doesn't affect the occurrence of another, known as independence.

Truth and Lies

Exploring the concept of truth and lies allows for understanding how probabilities of different statements work. When an individual like A says something, they either state the truth or lie. For A:

• Truth Probability: 60% (0.60)
• Lie Probability: 40% (1 - 0.60 = 0.40)

Similarly, another person, B, might also have stats on truth and lies:

• Truth Probability: 80% (0.80)
• Lie Probability: 20% (1 - 0.80 = 0.20)

This concept ties directly into scenarios where we consider if both parties say the same thing, regardless if it's the truth or a lie. Such concepts are crucial when analyzing facets of probability where failure and success coexist.

Event Probability

Event probability is the basic idea of finding out how likely an event is to occur. It involves straightforward calculations using known percentages. When determining outcomes such as whether A and B will say the same thing, we consider separate scenarios:

• Both tell the truth: Calculated by multiplying the individual truth probabilities.
• Both lie: Calculated by multiplying the individual lie probabilities.

Event probability typically uses simple arithmetic to paint a clear picture of potential outcomes. These calculations then allow for more complicated scenarios where multiple smaller events influence a larger picture, aiding in understanding holistic event outcomes.

Basic Multiplication Rule

The basic multiplication rule is a core principle in probability that aids in calculating the joint probability of independent events. The rule states that the probability of both events A and B occurring is the product of their individual probabilities.
In our case, calculating the probability of A and B both telling the truth would be:\[P(A \text{ truth}) \times P(B \text{ truth}) = 0.60 \times 0.80 = 0.48\]Additionally, for both A and B to lie, we compute:\[P(A \text{ lie}) \times P(B \text{ lie}) = 0.40 \times 0.20 = 0.08\]Once these individual events are calculated, they can be combined to derive further probabilities, such as the chance of both saying the same thing - whether it's both truths or both lies. This rule is often utilized in various scenarios to make sense of complex problems.