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\textit{Tabulated values of }${\Phi}(\cdot)$\textit{, the cumulative distribution function of a standard normal variable, should not be used in this question.} 
 Henry the  commuter lives in Cambridge and his working day starts at his office in London at 0900. He catches the 0715 train to King's Cross with probability $p$, or the 0720 to Liverpool Street with probability $1-p$. Measured in minutes, journey times for the first train are $N(55,25)$ and for the second are $N(65,16)$. Journey times from King's Cross and Liverpool Street to his office are $N(30,144)$ and $N(25,9)$, respectively. Show that Henry is more likely to be late for work if he catches the first train.
Henry makes $M$ journeys, where $M$ is large. Writing $A$ for $1-{\Phi}(20/13)$  and $B$ for $1-{\Phi}(2)$, find, in terms of $A$, $B$, $M$ and $p$, the expected number, $L$, of times that Henry will be late and show that for all possible values of $p$,
$$BM \le L \le AM.$$ 
Henry noted that in 3/5 of the occasions when he was late, he had caught the King's Cross train. Obtain an estimate of $p$ in terms of $A$ and $B$.
[A random variable is said to be $N\left({{\mu}, {\sigma}^2}\right)$ if it has a normal distribution with mean ${\mu}$ and variance ${\sigma}^2$.]

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