ONLY 326 b and 328 After a computer virus entered the system

ONLY 3.26 (b) and 3.28

After a computer virus entered the system, a computer manager checks the condition of all important files. She knows that each file has probability 0.2 to be damaged by the virus, independently of other files. Compute the probability that at least 5 of the first 20 files are damaged. Compute the probability that the manager has to check at least 6 files in order to find 3 undamaged files. 3.27. Messages arrive at an electronic message center at random times, with an average of 9 messages per hour. What is the probability of receiving at least five messages during the next hour? What is the probability of receiving exactly five messages during the next hour? The number of received electronic messages has Poisson distribution with some parameter lambda. Using Chebyshev inequality, show that the probability of receiving more than 4 lambda messages does not exceed 1/(9 lambda).

Solution

a)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.2      
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   4   ) =    0.629648264
          
Thus, the probability of at least   5   successes is  
          
P(at least   5   ) =    0.370351736 [answer]

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b)

Let getting a file \"damaged by virus\" be a success.

Here, we have P(at least 3 non-successes, 3 successes) = 1 - P(at most 2 non-successes, 3 successes)

Using a cumulative binomial table or technology,

P(at most 2 non-successes, 3 successes) = 0.05792

Thus,

P(at least 3 non-successes, 3 successes) = 0.94208 [answer]

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