A wildlife biologist examines frogs for a genetic trait he s

A wildlife biologist examines frogs for a genetic trait he suspects may be linked to sensitivity to industrial toxins in the environment. Previous research had established that this trait is usually found in 1 of every 8 frogs. He collects and examines a dozen frogs. If the frequency of the trait has not changed, what\'s the probability he finds the trait in none of the 12 frogs? at least 2 frogs? 3 or 4 frogs? no more than 4 frogs?

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 =    12      
p = the probability of a success =    0.125      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.201417238 [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 =    12      
p = the probability of a success =    0.125      
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.546703932
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.453296068 [ANSWER]

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c)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    3      
x2 =    4      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    12      
p = the probability of a success =    0.125      
          
Then          
          
P(at most    2   ) =    0.81800062
P(at most    4   ) =    0.988714522
          
Thus,          
          
P(between x1 and x2) =    0.170713902   [ANSWER]

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D)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    12      
p = the probability of a success =    0.125      
x = the maximum number of successes =    4      
          
Then the cumulative probability is          
          
P(at most   4   ) =    0.988714522 [ANSWER]