A company is looking at four high end motion control system designs. The probabi

Question

A company is looking at four high end motion control system designs. The probability of successful operation for each design is 0.40, independent of the results for any other system. The cost of each system is $200,000. Each design that is successful with be worth $600,000

(A) What is the probability that one or more system designs will be operating successfully?

(B) What is the expected number of system designs that would be successful?

(C) What is the expected gain for the company?

(D) What will be the gain if only one of the system designs is successful?

(E) Considering all possible results, what is the probability of a loss rather than a gain?

(F) What is the standard deviation of the number of successes?

Explanation / Answer

a)

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 =    4      
p = the probability of a success =    0.4      
x = our critical value of successes =    1      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   0   ) =    0.1296
          
Thus, the probability of at least   1   successes is  
          
P(at least   1   ) =    0.8704 [ANSWER]

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b)

E(x) = n p = 4*0.4 = 1.6 [ANSWER]

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c)

Here, if X = the number of successful operations,

Gain = 600000x - 200000*4

or

Gain = 600000x - 800000

Thus,

E(Gain) = 600000E(x) - 800000 = 600000*1.6 - 800000

E(Gain) = $160,000 [ANSWER]

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D)

Here, if X = the number of successful operations,

Gain = 600000x - 800000

If x = 1,

Gain = 600000(1) - 800000

Gain = -200,000 [ANSWER, it's actually a loss!]

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