You may use technology to perform some of the following calculations. Be sure to

Question

You may use technology to perform some of the following calculations. Be sure to justify all calculations and list any assumption that are made. Suppose a fair die is tossed 3 million times.Each time,the number 1,2,3,4,5, or 6 is recorded (it's a six-sided die).Let X be the number of times a 5 is recorded. What formula would you use to calculate p(x=5)? Explain the challenges of calculating this with a scientific calculator. Calculate the probability that P(X gretter than equal to 499000).You may calculate exactly or give a good approximation.

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)

Thus, if n = 3000000, x = 5, p = 1/6

P(n, x) = 3000000C5 (1/6)^5 (1 - 1/6)^(3000000 - 5)

However, the challenge is that the exponents are too high, and the factorials involved in 3000000C5 are too high as well.

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b)

We first get the z score for the critical value:          
          
x = critical value =    498999.5      
u = mean = np =    500000      
          
s = standard deviation = sqrt(np(1-p)) =    645.4972244      
          
Thus, the corresponding z score is          
          
z = (x-u)/s =    -1.549967935      
          
Thus, the left tailed area is          
          
P(z >   -1.549967935   ) =    0.939425394 [ANSWER]

          

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