In a certain rural area of New York State 80 of drivers use

In a certain rural area of New York State, 80% of drivers use chains on the car tires for winter driving. A random sample of 12 drivers is taken. You are interested in the number of drivers in the sample that use chains while driving. In words, define the random Variable X. What kind of distribution does X have and what are its parameters, specific to this problem? Find the probability that at most 8 drivers in the sample use chains. Find the probability that less than 8 drivers in the sample use chains. Find the probability that at least 8 drivers in the sample use chains. Find the probability that between 7 and 11 (inclusive) drivers in the sample use chains.

Solution

a)

X = the number of drivers who use chains while driving out of the 12 sampled

It will have a binomial distrbution, as the number of trials is fixed, and the probability of success is fixed.

b)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    12      
p = the probability of a success =    0.8      
x = the maximum number of successes =    8      
          
Then the cumulative probability is          
          
P(at most   8   ) =    0.20543105 [answer]

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c)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    12      
p = the probability of a success =    0.8      
x = our critical value of successes =    8      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   7   ) =    0.0725555
          
Which is also          
          
P(fewer than   8   ) =    0.0725555 [ANSWER]

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d)

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 =    12      
p = the probability of a success =    0.8      
x = our critical value of successes =    8      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   7   ) =    0.0725555
          
Thus, the probability of at least   8   successes is  
          
P(at least   8   ) =    0.9274445 [answer]

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e)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    7      
x2 =    11      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    12      
p = the probability of a success =    0.8      
          
Then          
          
P(at most    6   ) =    0.019405279
P(at most    11   ) =    0.931280523
          
Thus,          
          
P(between x1 and x2) =    0.911875244   [answer]