A total of 16 mice are sent down a maze, one by one. From previous experience, i

Question

A total of 16 mice are sent down a maze, one by one. From previous experience, it is believed that the probability a mouse turns right is .38. Suppose their turning pattern follows a binomial distribution. Use the PDF or the CDF command to help answer each of the following questions.

a) what is the probability that exactly 8 of the 16 mice turn right?

b) what is the probability that 8 or fewer of the 16 mice turn right?

c) what is the probability that 8 or more turn right?

d) what is the probability that more than 3, but fewer than 10 turn right?

e) what is the probability that exactly 10 turn left?

Explanation / Answer

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 =    16      
p = the probability of a success =    0.38      
x = the number of successes =    8      
          
Thus, the probability is          
          
P (    8   ) =    0.122174575 [ANSWER]

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b)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    16      
p = the probability of a success =    0.38      
x = the maximum number of successes =    8      
          
Then the cumulative probability is          
          
P(at most   8   ) =    0.892371906 [ANSWER]

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c)

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 =    16      
p = the probability of a success =    0.38      
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   ) =    0.770197331
          
Thus, the probability of at least   8   successes is  
          
P(at least   8   ) =    0.229802669 [ANSWER]

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d)

That means 4 to 9 inclusive.

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    4      
x2 =    9      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    16      
p = the probability of a success =    0.38      
          
Then          
          
P(at most    3   ) =    0.088106891
P(at most    9   ) =    0.958932965
          
Thus,          
          
P(between x1 and x2) =    0.870826074   [ANSWER]

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e)

Note that p(left) = 1 - 0.38 = 0.62.

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 =    16      
p = the probability of a success =    0.62      
x = the number of successes =    10      
          
Thus, the probability is          
          
P (    10   ) =    0.202368326 [ANSWER]
     

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