Question

A signal which can be green or red with probability \(\frac{4}{5}\) and \(\frac{1}{5}\) respectively, is received by station \(A\) and then transmitted to station \(B .\) The probability of each station receiving the signal correctly is \(\frac{3}{4} .\) If the signal received at station \(B\) is green, then the probability that the original signal was green is A) \(\frac{3}{5}\) B) \(\frac{6}{7}\) (C) \(\frac{20}{23}\) D) \(\frac{9}{20}\)

Short answer

The correct answer is B) \frac{6}{7}.

Step-by-step solution

Define the Probabilities

Let the event that the original signal is green be G, and the event that the original signal is red be R. Let A and B be the events that station A and station B receive the signal correctly, respectively. We have: P(G) = 4/5, P(R) = 1/5, P(A) = P(B) = 3/4.

Calculate the Probabilities of Correct Transmission

Calculate the probability that station A transmits a green signal correctly to station B, which is P(G and A) = P(G) * P(A) = (4/5) * (3/4). Similarly, calculate P(R and A) = P(R) * P(A) = (1/5) * (3/4).

Determine the Probability of B Receiving Green

Next, we need to find the probability that B receives a green signal, which could either be because the original was green and correctly transmitted, or it was red and incorrectly transmitted as green. So, we need to add the probability of A transmitting green correctly and the probability of A incorrectly transmitting red as green. This can be calculated as P(G and A) * P(B given G and A) + P(R and A) * P(G given R and A).

Apply Bayes' Theorem

We need to find the probability that the signal was green given that station B received green, which is P(G | B). Apply Bayes' Theorem which tells us that P(G | B) = [P(B | G and A) * P(G)] / P(B).

Compute the Probability That B Receives Green Given Green and A

P(B | G and A) is simply the probability of B receiving the signal correctly, so P(B | G and A) = P(B).

Compute the Probability B Receives Green

Use the probability that B receives green correctly from a green signal and incorrectly from a red signal to compute P(B).

Put It All Together

Finally, substitute the values from the previous steps into Bayes' Theorem and solve for P(G | B).

Key concepts

Probability Theory

Probability theory is the branch of mathematics that deals with the analysis of random events. The fundamental components include outcomes, events, and the probability of events. An outcome is a result of a single trial of an experiment, while an event is a set of outcomes for which we can assign a probability. The probability, denoted as P(event), is a measure of the likelihood that the event will occur, and ranges between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

In the context of the given problem, probability theory provides a framework for quantifying how likely it is that particular events occur, such as the original signal being green or red, and whether it will be received correctly by the stations involved in the communication process. The problem begins by defining simple probabilities for these events and then progresses to combine them to address more complex questions about the transmission of the signal.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. This concept is expressed as P(A | B), read as 'the probability of event A given event B.' It is crucial in situations where we want to update our probability assessments in light of new information or events.

In our exercise, conditional probability comes into play when determining the likelihood that the original signal was green, given that station B received a green signal. This essentially incorporates the idea of dependence between these two events. The solution involves computing such conditional probabilities and links them through Bayes' Theorem, which is essential for updating probabilities based on new information.

Correct Signal Transmission

Correct signal transmission is a particular scenario in communication systems where a signal sent from one point is received accurately at another point. In terms of probability, we assess the reliability of transmission lines or communication stations by assigning a probability to the event of a signal being transmitted without error. In real-world applications, such as digital communications, ensuring high probability of correct signal transmission is vital for data integrity and overall system performance.

When addressing problems in correct signal transmission using probability theory and conditional probability, we often seek to understand the likelihood of receiving the intended signal amidst the possibility of transmission errors. The probability of correct transmission and potential errors are combined to derive the overall probability of receiving a certain signal at the destination, which is precisely what's illustrated in the step-by-step solution of our exercise.