In San Francisco 30 of workers take public transportation da

In San Francisco, 30% of workers take public transportation daily (USA Today, December 21,2005).

In a sample of 15 workers, what is the probability that exactly three workers take public transportation daily (to 4 decimals)?
  

In a sample of 15 workers, what is the probability that at least three workers take public transportation daily (to 4 decimals)?

Solution

a)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    15      
p = the probability of a success =    0.3      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.170040213 [ANSWER]

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b)

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 =    15      
p = the probability of a success =    0.3      
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.126827715
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.873172285 [ANSWER]