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Problem 3085
3085
In a certain country, 60% of mobile phones sold are made by Company A, 35% are made by Company B, and 5% are made by other companies.
A random sample of n mobile phones sold is chosen. The probability that at least one of these phones is made by Company B is more than 0.98. Find the least possible value of n.
Solution
The probability that a phone is not made by Company B is 0.65 (since 35% are made by Company B).
The probability that none of the n phones is made by Company B is given by
\((0.65)^n\).
We want the probability that at least one phone is made by Company B to be more than 0.98, so:
\(1 - (0.65)^n > 0.98\)
\((0.65)^n < 0.02\)
Taking logarithms on both sides:
\(n \log(0.65) < \log(0.02)\)
\(n > \frac{\log(0.02)}{\log(0.65)}\)
Calculating the right-hand side gives \(n > 9.08\).
Since n must be an integer, the smallest possible value is \(n = 10\).