Suppose at the shooting range you have a probability of hitt

Suppose at the shooting range you have a probability of hitting the bullseye on your target about 25%.

(a)what is the probability of hitting the bullseye exactly 5 times in 20 shots?
(b)what is the approximate probability of hitting 121 bullseyes or less in 550 shots?

please show work, thank you.

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 =    20      
p = the probability of a success =    0.25      
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.202331152 [ANSWER]

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B)

We first get the z score for the critical value:          
          
x = critical value =    121.5      
u = mean = np =    137.5      
          
s = standard deviation = sqrt(np(1-p)) =    10.15504801      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    -1.575571085      
          
Thus, the left tailed area is          
          
P(z <   -1.575571085   ) =    0.057562342 [ANSWER]