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Let A arrive in a party in the time interval [0,a] and B arrive at the same party in the time interval [0,b].

What will be the probability that both of them arrive at the same time ?

Note : Arrival time can be any real number in the interval ,not only integers .

I have no clue what to take as the total number of events or how to approach this problem . Please help .

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Zero. Assume that $a < b$ without loss of generality and pick some window of width $d$ in $[0,a]$. B will arrive in that window with probability $\frac{d}{b}$. Simply take the limit as $d \to 0$.

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  • $\begingroup$ But what if a=b then $\endgroup$ – Apoorv Jain Jun 6 '15 at 18:09
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    $\begingroup$ @user3650050 still zero. The difficulty here is that the "time at which $A$ arrives" is a set of measure zero w.r.t. the standard lebesgue measure. I.e. the "width" of the interval zero. It is like asking "If I pick two real numbers between 0 and 1, what is the probability that I picked the same number both times." While it is not impossible to pick the same number twice in a row, it is so unlikely that we still say that it occurs with probability zero. Afterall, after picking the first number, there is a $\frac{1}{m([0,1])} = \frac{1}{c} \approx 0$ chance of repetition. $\endgroup$ – JMoravitz Jun 6 '15 at 21:31
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    $\begingroup$ The probability will be positive however if there are a finite number of points of time where $A$ can arrive (i.e. $A$ and $B$ can only arrive at exactly integer multiples of 15 seconds on the clock) or if you allow them to have some sort of nonzero measure (i.e. if you ask "what is the probability that $A$ and $B$ arrive at the party during the same minute?") $\endgroup$ – JMoravitz Jun 6 '15 at 21:33

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