Question

Problem 1: Three machines in the Olde Factory, M1, M2, and M3 are each capable of running at one of only two rates: low (L) and high (H). The H setting affords the greatest productivity but at an increased maintenance cost while the L setting produces just the opposite result. Automatic control software continually adjust the machine’s two rates in an attempt to balance productivity with maintenance costs. At any given time the configuration of the three machines is designated by writing a sequence of three letters, L or H, that designate by their position the rates of machines 1, 2, and 3. For example, the configuration LHH indicates that machine 1 is operating on the low setting and machines 2 and 3 are operating on the high setting. The following have been noted when observing the machines at a randomly chosen time: P[Exactly 1 Machine in state H] = 0.56 P[HHL] = 0.02 P[HLH] = 0.04 P[LHH] = 0.01 P[M1 = H] = 0.56 P[M2 = H] = 0.37 P[M3 = H] = 0.31 Part 1a: Draw the appropriate Venn or Tree diagram that describes this situation. If Venn, make sure all atomic regions have values. If Tree, make sure all leaf probabilities are computed. Part 1b: What is the probability that all machines are operating at the same rate (all H or all L?) Part 1c: Which combination of rates is observed most often? Part 1d: If two of the machines are operating at the L rate, what is the probability that the third machine is also operating at the L rate?

          Problem 1: Three machines in the Olde Factory, M1, M2, and M3 are each capable of running at one of only two rates: low (L) and high (H). The H setting affords the greatest productivity but at an increased maintenance cost while the L setting produces just the opposite result. Automatic control software continually adjust the machine’s two rates in an attempt to balance productivity with maintenance costs. At any given time the configuration of the three machines is designated by writing a sequence of three letters, L or H, that designate by their position the rates of machines 1, 2, and 3. For example, the configuration LHH indicates that machine 1 is operating on the low setting and machines 2 and 3 are operating on the high setting.

The following have been noted when observing the machines at a randomly chosen time:

P[Exactly 1 Machine in state H] = 0.56
P[HHL] = 0.02 P[HLH] = 0.04 P[LHH] = 0.01
P[M1 = H] = 0.56 P[M2 = H] = 0.37 P[M3 = H] = 0.31

Part 1a: Draw the appropriate Venn or Tree diagram that describes this situation. If Venn, make sure all atomic regions have values. If Tree, make sure all leaf probabilities are computed.
Part 1b: What is the probability that all machines are operating at the same rate (all H or all L?)
Part 1c: Which combination of rates is observed most often?
Part 1d: If two of the machines are operating at the L rate, what is the probability that the third machine is also operating at the L rate?

Problem 1: Three machines in the Olde Factory, M1, M2, and M3 are each capable of running at one of only two rates: low (L) and high (H). The H setting affords the greatest productivity but at an increased maintenance cost while the L setting produces just the opposite result. Automatic control software continually adjust the machine’s two rates in an attempt to balance productivity with maintenance costs. At any given time the configuration of the three machines is designated by writing a sequence of three letters, L or H, that designate by their position the rates of machines 1, 2, and 3. For example, the configuration LHH indicates that machine 1 is operating on the low setting and machines 2 and 3 are operating on the high setting.

The following have been noted when observing the machines at a randomly chosen time:

P[Exactly 1 Machine in state H] = 0.56
P[HHL] = 0.02 P[HLH] = 0.04 P[LHH] = 0.01
P[M1 = H] = 0.56 P[M2 = H] = 0.37 P[M3 = H] = 0.31

Part 1a: Draw the appropriate Venn or Tree diagram that describes this situation. If Venn, make sure all atomic regions have values. If Tree, make sure all leaf probabilities are computed.
Part 1b: What is the probability that all machines are operating at the same rate (all H or all L?)
Part 1c: Which combination of rates is observed most often?
Part 1d: If two of the machines are operating at the L rate, what is the probability that the third machine is also operating at the L rate?

Added by Tony C.

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