26. A shopping mall has three automated teller machines (ATMs). Because the mach

Question

26. A shopping mall has three automated teller machines (ATMs). Because the machines recelve heavy use, they sometimes stop working and need to be repaired. Let the random variable X represent the mumber that are working when the mall opens on a randomly selected day. The table shows the probability distribution ofx. of ATMs | 0 # of ATMs working when the mall opens Probability | 1 0.15 0.21 0.40 0.24 (a) What is the probablity that at least one ATM is working when the mall opens? (b) What is the expected value of the number of ATMs that are working when the mall opens? (c) What is the probability that all three ATMs are working when the mall opens, gliven that at least one ATM is working? (d) Given that at least one ATM is working when the mall opens, would the expected value of the number of ATMs that are working be less than, equal to, or greater than the expected value from part (b)? Explain

Explanation / Answer

Question 26:

a) Probability that at least one ATM is working is computed here as:

P( X >= 1) = 1 - P(X = 0) = 1 - 0.15 = 0.85

Therefore 0.85 is the required probability here.

b) The expected value here is computed as:

E(X) = 0*0.15 + 1*0.21 + 2*0.4 + 3*0.24 = 1.73

Therefore 1.73 is the expected number of ATMs here.

c) Given that at least one ATM is working, probability that all three ATMs are working when the mall opens is computed here as:

= Probability that all three ATMs are open / Probability that at least one ATM is open

= 0.24 / 0.85

= 0.2824

Therefore 0.2824 is the required probability here.

d) Given that at least one ATM is working when the mall opens, the expected value of the number of ATMs that are working would be more than the expected value from part b) because here we would not be taking the case of P(X = 0) here.

Therefore greater than is the correct answer here.

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