U1 arrival time of A. U2 arrival time of B
Red area shows A’s distribution and Blue area shows B’s distribution.
P(U1>U2) = P(U1>U2 | U1>9, U2<10 ) P(U1 > 9, U2<10 )
Uniform conditioned to be in an interval is uniform. U1,U2 are independent, so conditioning of one has no affect on the other.
Therefore,
P(U1>U2) = P( U′1 > U′2 ) P(U1 > 9 ) P( U2 < 10 )
where U′1,U′2 are independent and uniform on [9,10]. (how i interpret this is, the probability of anything happening in this area(where red and blue intersect) is ½ which includes U’1 > U’2 since it is uniformly distributed)
P( U′1 > U′2 ) = ½
P(U1 > 9 ) = (10-9)*(1/2) = ½
P(U1 > 9 ) = (11-10)*(1/2) = ½
Evaluating the three terms, we obtain:
1/2×1/2×1/2=1/8.
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