The Overloaded Cell Network
A small town's cell network has three towers. Tower A covers 40% of the town's area, Tower B covers 35%, and Tower C covers 25%. The overlap regions are small but real: A and B overlap on 5% of the town's area, B and C overlap on 4%, and A and C overlap on 3%. No area is covered by all three towers. The town council wants to know how much of the town has no coverage at all, and a reporter claims that 'since the towers cover 100% of the town between them, everyone has service.' A separate survey found that 12% of residents report dropped calls, but the survey only reached people with landlines, who make up 60% of the population.
1. Using the coverage percentages and overlaps, determine what fraction of the town has no cell coverage. Then evaluate the reporter's claim.
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Add the individual coverages: 40% + 35% + 25% = 100%. Subtract each overlap once because that area was counted twice: 5% + 4% + 3% = 12%. Since no area is covered by all three, no triple-overlap correction is needed. Covered area = 100% − 12% = 88%. So 12% of the town has no coverage. The reporter's claim is wrong: the towers' individual coverages sum to 100%, but because they overlap, the union covers only 88%. Summing percentages of a whole is only valid when the groups are disjoint.
2. The dropped-call survey reached only landline households (60% of the population). Explain why the 12% dropped-call figure cannot simply be reported as the townwide rate, and describe what additional information would be needed to estimate the true rate.
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The survey sample is not representative: landline-only households differ systematically from the 40% who have no landline (for example, they may be older or more likely to live in areas with better or worse infrastructure). The 12% is a rate for landline households only, not the whole town. To estimate the townwide rate, you would need the dropped-call rate among non-landline households as well, then combine them weighted by population share: (0.60 × 12%) + (0.40 × r), where r is the unknown rate for the other group. Without r, the best you can say is that the townwide rate equals 7.2% plus 0.4r, which could be anywhere from 7.2% (if r = 0) upward. This is a sampling-bias issue, not a math error.