Hi! Could you please help me with this? over a long period of time in a latge co

Question

Hi! Could you please help me with this?

over a long period of time in a latge corporation, 10% of all sales trainees are rated as outstanding, 75% are rated as excellent/good, 10% are rated as satisfactory, and 5% are considered unsatisfactory. find the following probabilities for a sample of 10 traniees selected at a random:

a. two are rated as as outsanding

b. two or more are rated as outstanding

c. eight of the ten rated either outstanding or excellent/good.

d. none of the traniees is rated as unsatistactory.

Thanks!

Explanation / Answer

a)

Note that P(outstanding) = 0.10.

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 =    10      
p = the probability of a success =    0.1      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.193710245 [ANSWER]

********************

b)

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 =    10      
p = the probability of a success =    0.1      
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   ) =    0.736098929
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.263901071 [ANSWER]

**************************

c)

Note that

P(outstanding or excellent) = 0.10 + 0.75 = 0.85.

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 =    10      
p = the probability of a success =    0.85      
x = the number of successes =    8      
          
Thus, the probability is          
          
P (    8   ) =    0.275896657 [ANSWER]

******************

d)

Note that

P(unsatisfactory) = 0.10.

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 =    10      
p = the probability of a success =    0.1      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.34867844 [ANSWER]

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