5) A system of 32 remotely operated radar stations is monitoring the airspace ov

Question

5) A system of 32 remotely operated radar stations is monitoring the airspace over the North-West Territories. The Central Command has experienced difficulties with monitoring which are believed to be due to technical problems at 3 stations. However, it is not known which of the 32 stations have problems. Four stations are randomly selected at a time, and a diagnostic test is run. What is the probability that of those four stations, a) at least one has a problem? b) exactly one has a problem? c) exactly three have a problem? d) none has a problem?

Explanation / Answer

4 stations from 32 stations can be selected in 32C4 ways

(d) None has a problem

All four sample stations selected from non-defective stations i.e. 32-3 = 29 stations. This can be done in 29C4 ways

Hence requide probability P(D) = 29C4/32C4 = 0.66

(a) At least one has problem
Required probability = 1 - P(D) = 1 - 0.66 = 0.34

(b) Exactly one has problem
One station from 3 defective stations can be selected in 3C1 ways
Remaining 3 non-defective stations can be selected in 29C3 ways

Hence required probability = 3C1*29C3/32C4 = 0.3048

(c) Exactly 3 have problem

Three stations from 3 defective stations can be selected in 3C3 = 1 way
Remaining 1 non-defective stations can be selected in 29C1 ways

Hence required probability = 3C3*29C1/32C4 = 0.000806

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