Section B ( 53 marks )
10 . At 8 am on any school day , Stephen walks from his apartment to the school bus stop . The time T minutes needed for Stephen to walk to the school bus stop follows a normal distribution such that
2
T ~ N ( 16 , 5 ) .
There are two school buses which will depart at 8:12 am and 8 : 24 am respectively , in every school day morning . Stephen will take the first bus if he arrives at or before 8:12 am .
( a ) Find the probability that Stephen can arrive at the school bus stop before the second school bus departs .
The time U minutes needed for a school bus to travel from the school bus stop to school follows a normal distribution such that
[ 2 ]
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U ~ N ( � , 7 ) . It is given that U and T are independent , and 99.494 % of the school buses take at most 48 minutes to travel from the school bus stop to school .
( b ) Find � .
[ 3 ] In order to be marked as on time , Stephen needs to take any one of the buses and arrive at school by 9 am .
( c ) Show that , correct to five significant figures , for all school buses departing at 8 : 24 am , 80.439 % of them will arrive at school on time .
[ 2 ] ( d ) Hence , find the probability that Stephen will not arrive at school on time .
[ 5 ]
There are twenty school days in February 2021 .
( e ) Find the expected number of school days that Stephen will not arrive at school on time in February 2021 .
[ 2 ]
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