Memory module consists of 9 chips. The device is designed with redundancy so that it works even if one of its chips is defective. Each chip contains transistors and function properly if all its transistors work. A transistor works with probability p independent of each other transistor.(a) What is the probability P[C] that a chip works? (b) What is the probability P[M] that the memory module works? (c) If p = 0.999, what is the maximum number of transistors per chip n that produces P[M] ≥ 0.9 (a 90% success probability for the memory module)? (d) If the memory module can tolerate two defective chips, what is the maximum number of transistors per chip n that produces P[M] ≥ 0.9?

a)

b)

c) n=62

d) n=138

Step-by-step explanation:

Note: "Each chip contains n transistors"

a) A chip needs all n transistor working to function correctly. If p is the probability that a transistor is working ok, then:

b) The memory module works with when even one of the chips is defective. It means it works either if 8 chips or 9 chips are ok. The probability of the chips failing is independent of each other.

We can calculate this as a binomial distribution problem, with n=9 and k≥8:

c)

This equation was solved graphically and the result is that the maximum number of chips to have a reliability of the memory module equal or bigger than 0.9 is 62 transistors per chip. See picture attached.

d) If the memoty module tolerates 2 defective chips:

We again calculate numerically and graphically and determine that the maximum number of transistor per chip in this conditions is n=138. See graph attached.

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step-by-step explanation: