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A lot of five identical batteries is life tested. The probability assignment is assumed to be $$ P(A)=\int_A\ (1/\lambda) * e^{-x/\lambda}dx\ $$

for any event A$\sqsubseteq[0,\infty)$, where $\lambda$>0 is a known constant. Thus the probability that a battery fails after time t is given by $$ P(t,\infty) = \int_t^\infty (1/\lambda) * e^{-x/\lambda}dx\,, t\geq 0 $$

If the times to failure of the batteries are independent, what is the probability that at least one battery will be operating after $t_0$ hours?

My Way of Solving This Problem: $$ $$ Let F be the number of batteries that fail after time $t_0$ & n=5 is the number of batteries in the lot. Then we are interested in $P(F\leq4)=P$(at least one battery did not fail after time $t_0$). So I used a binomial distribution where success was described to be a battery failing after time $t_0$. $$ P(F\leq4)=1-P(F=5) $$ $$ =1-{5 \choose 5}p^5*\bigl(1-p\bigr)^0 $$ $$ =1-p^5 $$ $$ =1-\bigl(\int_{t_0}^\infty (1/\lambda) * e^{-x/\lambda}dx\bigr)^5 $$ $$ =1-e^{-5t_0/\lambda} $$

I have been told that this answer is wrong but I do not understand why my logic is flawed. Please help me find out why my logic is flawed.

kjetil b halvorsen
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J.Doe
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  • I realize that there have been solutions to this question posted before but from what I can tell they seem to be contradictory. @Glen_b seems to agree with my approach of doing things when he responded to http://stats.stackexchange.com/questions/176479/probabiity-at-least-one-survives-under-independence. – J.Doe Sep 29 '16 at 22:05
  • However @Max seems to disagree with my method when he responded to http://math.stackexchange.com/questions/1927490/finding-probability-on-operation-time-of-batteries – J.Doe Sep 29 '16 at 22:06

1 Answers1

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Please don't crosspost to other sites. I think this is an issue of wording. The question is "what is the probability that at least one battery will be operating after $t_0$ hours". Your interpretation (which I think is reasonable) is the chance of at least one battery surviving to infinity. Perhaps the question instead meant "what's the chance at least one battery didn't fail in time interval $[0,t_0]$." This would give you the reverse: $1-p$ in your probability.

Alex R.
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  • Thank you for your response. I thought that it was okay to put a link to posts made on stackexchange.com – J.Doe Sep 29 '16 at 22:20
  • You see I too believe it is an issue with wording. Though the lack of clarity that comes from the phrasing of the problem makes it difficult, I think I understand your interpretation. – J.Doe Sep 29 '16 at 22:25