“SmartB” owns a phone battery assembly line which provide phone battery to a mobile company. It is known that 4% of the batteries produced by “SmartB” are defective.
The production cost of a phone battery is ＄100 and the selling price to the mobile company is ＄500. When the phone battery is found defective within a year, it could be sent back for repairment, which costs “SmartB” ＄70. Suppose all defective phone battery would be reported by the customer within a year and would be sent back for repairment. Present the probability distribution function of the profit gained by “SmartB” from a phone battery. (Remark: Profit = revenue – cost)
My approach:
Let N be the number of phone produced. P(N) = 500N - (100N + 70 x Bin(N, 0.04)) = 400N - 70 x Bin(N, 0.04)
Is it correct? I think my answer are incorrect as it should be a value but not contain variable
The probability that $k$ (out of n) phone batteries are faulty is $P(X=k)=\binom{n}k\cdot 0.04^k\cdot 0.96^{n-k}$. A phone battery which is not faulty gains $\$500-\$100=\$400$. I agree. A phone battery which is faulty gains $\$500-\$100-\$70=\$330$. Therefore the profit for $k$ faulty and $(n-k)$ non-faulty phone batteries is $k\cdot 330+(n-k)\cdot 400$. It remains to add the corresponding probability.