The probability that a communication system will have high fidelity is 0.81. The probability that the system will have both high fidelity and high selectivity is 0.18. The probability that a given system with high fidelity will have high selectivity is

This question asks us to calculate a conditional probability related to a communication system's performance characteristics: high fidelity and high selectivity.

Understanding Conditional Probability in Communication Systems

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. In this context, we want to find the probability of a system having high selectivity, given the condition that it already has high fidelity.

Given Probabilities

- Let F be the event that the communication system has high fidelity.
- Let S be the event that the communication system has high selectivity.
- We are given the probability of high fidelity: P(F) = 0.81.
- We are given the probability of both high fidelity and high selectivity: P(F and S) = 0.18.

The Formula for Conditional Probability

The formula to calculate the conditional probability of event S occurring given that event F has occurred is:

$$P(S|F)=\frac{P(F\text{ and }S)}{P(F)}$$

Step-by-Step Calculation

1. Identify the known values: - $P(F\text{ and }S)=0.18$ - $P(F)=0.81$
2. Apply the conditional probability formula: Substitute the known values into the formula: $$P(S|F)=\frac{0.18}{0.81}$$
3. Perform the division: Calculate the value: $$P(S|F)=\frac{18}{81}$$ To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 9: $$P(S|F)=\frac{18÷9}{81÷9}=\frac{2}{9}$$ Now, convert the fraction to a decimal: $$\frac{2}{9}\approx \mathrm{0.2222...}$$
4. Conclusion: The probability that a given system with high fidelity will also have high selectivity is approximately 0.222.

Final Answer Check

Comparing our calculated result with the given options, we find that 0.222 matches the third option.