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The lifetime of two brands of bulbs $X$ and $Y$ are exponentially distributed with the mean life of $100$ hours. Bulb $X$ is switched on $15$ hours after Bulb $Y$ has been switched on. The probability that bulb $X$ fails before Bulb $Y$ is?

The options are:

$(A). \frac{15}{100}$

$(B). \frac{1}{2}$

$(C).\frac{85}{100}$

$(D). 0$

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    Hint: exponential distribution is [memoryless](https://en.wikipedia.org/wiki/Exponential_distribution#Memorylessness). Bulb $Y$ will not suffer under the fact that it is allready burning for $15$ hours when $X$ is switched on. Secondly you are dealing with equal distributions. – 2017-01-27
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    So we we'll have to find cdf? P(X – 2017-01-27
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    Do we know that bulb $Y$ is still functioning the moment bulb $X$ is switched on? – 2017-01-27
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    It's not given in the question. But I think it would be. – 2017-01-27

2 Answers 2

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If bulb $Y$ might fail before bulb $X$ is switched on,

\begin{align} &\text{ probability that bulb $X$ fails before bulb } Y \\ &= P(Y \geq 15) P(X

If we are given that bulb $Y$ is still working when bulb $X$ is switched on \begin{align} &\text{ probability that bulb $X$ fails before bulb } Y \text{ given that bulb } Y \text{ was functioning when bulb } X \text{was switched on} \\ &= P(X

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    Actually the options (rational and including $\frac12$) indicate that the condition $Y\geq15$ is abusively not mentioned. Secondly in the title the OP asks for $P(X – 2017-01-27
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    +1, interpreting the question as asking for $P(Y>X+15 | Y>15)$ is the only way I can see to make it fit the given answers. – 2017-01-27
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Edit: (Bulb X is switched on 15 hours after bulb Y)

You must partition on the 15 hour.   Bulb-X may fail before bulb-Y, only if bulb-Y does not fail before bulb X is switched on. $$\mathsf P(X

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    Concerning LHS: since $X,Y$ are iid and continuous, it is immediate that $P(X$X$ and $Y$. – 2017-01-27
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    $X$ and $Y$ are not iid because bulb $X$ is switched on 15 hours later. $Y$ and $X-15$ are iid. (But yes, I did have the switch on times mixed around) – 2017-01-27