PROBABILITY AND STATISTICS QUESTION Someone offers a prize t

PROBABILITY AND STATISTICS QUESTION!!!

Someone offers a prize to anyone who can get 125 or more 6\'s in 600 rolls of a fair die. What is the probability that if 300 people try, at least two will win prizes?

Solution

14.

We first get the z score for the critical value:          
          
x = critical value =    124.5      
u = mean = np =    100      
          
s = standard deviation = sqrt(np(1-p)) =    9.128709292      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    2.683840532      
          
Thus, the left tailed area is          
          
P(z >   2.683840532   ) =    0.003639091

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    1.0917273      
          
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   ) =    0.702059503
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.297940497 [ANSWER]