In a local cellular phone area, companies A and B account for 60% and 40% of the cellular phone market. 1% of the calls made with company A and 2% of the cellular calls with company B will have interference. If a cellular call is selected at random, the probability that it will have interference is:
0.014
Understanding Cellular Call Interference Probability
This problem requires us to calculate the overall probability that a randomly selected cellular call will experience interference. We are given the market share of two companies, A and B, and the conditional probability of interference given a call is made using each company's service.
Given Information:
- Market share of Company A, denoted as $P(A)$: 60% or 0.60
- Market share of Company B, denoted as $P(B)$: 40% or 0.40
- Probability of interference given the call is with Company A, denoted as $P(I|A)$: 1% or 0.01
- Probability of interference given the call is with Company B, denoted as $P(I|B)$: 2% or 0.02
We need to find the total probability of interference, denoted as $P(I)$.
Applying the Law of Total Probability
The situation described is a classic application of the Law of Total Probability. This law states that if an event (interference, I) can occur under several mutually exclusive and exhaustive conditions (using company A or company B), then the probability of the event is the sum of the probabilities of the event occurring under each condition.
In this case, the 'conditions' are which company's service is used for the call. Since companies A and B cover the entire market (60% + 40% = 100%), and a call is made using one or the other (mutually exclusive), they form a partition of the sample space.
The formula for the Law of Total Probability for this scenario is:
$$P(I) = P(I|A) \cdot P(A) + P(I|B) \cdot P(B)$$
Where:
- $P(I)$ is the total probability of interference.
- $P(I|A)$ is the probability of interference given Company A is used.
- $P(A)$ is the probability that Company A is used (its market share).
- $P(I|B)$ is the probability of interference given Company B is used.
- $P(B)$ is the probability that Company B is used (its market share).
Calculating the Probability of Interference
Now, we substitute the given values into the formula:
$$P(I) = (0.01 \cdot 0.60) + (0.02 \cdot 0.40)$$
First, calculate the probability of interference through Company A:
$$0.01 \cdot 0.60 = 0.006$$
This is the probability that a call is with Company A AND has interference.
Next, calculate the probability of interference through Company B:
$$0.02 \cdot 0.40 = 0.008$$
This is the probability that a call is with Company B AND has interference.
Finally, sum these probabilities to find the total probability of interference:
$$P(I) = 0.006 + 0.008$$
$$P(I) = 0.014$$
Therefore, the probability that a randomly selected cellular call will have interference is 0.014.
Summary of Calculation
| Component | Value | Calculation Contribution ($P(I|Company) \cdot P(Company)$) |
|---|---|---|
| Company A Market Share ($P(A)$) | 0.60 | $0.01 \cdot 0.60 = 0.006$ |
| Company A Interference Rate ($P(I|A)$) | 0.01 | |
| Company B Market Share ($P(B)$) | 0.40 | $0.02 \cdot 0.40 = 0.008$ |
| Company B Interference Rate ($P(I|B)$) | 0.02 | |
| Total Probability of Interference ($P(I)$) | $0.006 + 0.008 = 0.014$ |
Conclusion on Cellular Call Interference
Based on the market shares and conditional interference rates provided for companies A and B, the overall probability of a randomly selected cellular call experiencing interference is calculated to be 0.014.
Revision Table: Probability Concepts
| Concept | Description | Relevance to Cellular Interference Problem |
|---|---|---|
| Probability | A measure of the likelihood of an event occurring. Value between 0 and 1. | Used to quantify market share and interference rates. |
| Conditional Probability | The probability of an event occurring given that another event has already occurred, e.g., $P(I|A)$. | Given as the interference rate for each company. |
| Law of Total Probability | Used to find the total probability of an event by considering all possible mutually exclusive conditions under which it can occur. $P(E) = \sum P(E|C_i)P(C_i)$ for a partition $C_i$. | Applied directly to sum probabilities of interference from Company A and Company B. |
| Mutually Exclusive Events | Events that cannot occur at the same time. | A call is either with Company A or Company B, not both simultaneously. |
| Exhaustive Events | A set of events that includes all possible outcomes. | Companies A and B cover the entire cellular market (60% + 40% = 100%). |
Additional Information: Beyond Cellular Interference Probability
The principles used here, particularly the Law of Total Probability, are fundamental in many areas of probability and statistics. They are essential building blocks for more complex concepts like Bayes' Theorem, which is used to update probabilities based on new evidence. For example, if we knew a call had interference, Bayes' Theorem could help us find the probability that the call was made with Company A or Company B ($P(A|I)$ or $P(B|I)$).
Understanding how to combine conditional probabilities and overall probabilities is crucial in fields ranging from telecommunications (like analyzing signal quality) and finance (calculating risks) to medical diagnostics (interpreting test results) and machine learning (building predictive models). The cellular interference example provides a clear, tangible illustration of these core probabilistic ideas.
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