13 In this question you must show detailed reasoning.
The probability that Paul's train to work is late on any day is 0.15 , independently of other days.
- The number of days on which Paul's train to work is late during a 450-day period is denoted by the random variable \(Y\). Find a value of \(a\) such that \(\mathrm { P } ( Y > a ) \approx \frac { 1 } { 6 }\).
In the expansion of \(( 0.15 + 0.85 ) ^ { 50 }\), the terms involving \(0.15 ^ { r }\) and \(0.15 ^ { r + 1 }\) are denoted by \(T _ { r }\) and \(T _ { r + 1 }\) respectively.
- Show that \(\frac { T _ { r } } { T _ { r + 1 } } = \frac { 17 ( r + 1 ) } { 3 ( 50 - r ) }\).
- The number of days on which Paul's train to work is late during a 50-day period is modelled by the random variable \(X\).
(a) Find the values of \(r\) for which \(\mathrm { P } ( X = r ) \leqslant \mathrm { P } ( X = r + 1 )\).
(b) Hence find the most likely number of days on which the train will be late during a 50-day period.