Let X have a binominal distribution with n = 25, and p = 0.35). Find: The probab

Question

Let X have a binominal distribution with n = 25, and p = 0.35). Find: The probability of not more than 11 successes The probability of not less than 8 failures. P(3 lessthan equal to X lessthan 10) Let the Random Variable, X, follow the NEgatice Binomial probability distribution, with probability of success p = 0.55. what is the probability to make 7 trails in order to get 4 successes? Find the expected value of X Let X be a random variable with poisson distribution and parameter value of 4. Find the following: P (3 lessthan X lessthan equal to 7), with full details. P(at least 7 hits), with full details.

Explanation / Answer

2.

Binomial distribution:

I)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    25      
p = the probability of a success =    0.35      
x = the maximum number of successes =    11      
          
Then the cumulative probability is          
          
P(at most   11   ) =    0.874584234 [ANSWER]

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ii)

Note that p(fail) = 1 - 0.35 = 0.65.

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 =    25      
p = the probability of a success =    0.65      
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   ) =    0.000180779
          
Thus, the probability of at least   8   successes is  
          
P(at least   8   ) =    0.999819221 [ANSWER]

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iii)

Hence, between 3 and 9 successes inclusive.

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    3      
x2 =    9      
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    25      
p = the probability of a success =    0.35      
          
Then          
          
P(at most    2   ) =    0.002133337
P(at most    9   ) =    0.630308981
          
Thus,          
          
P(between x1 and x2) =    0.628175644   [ANSWER]  

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