I invite \(100\) special friends to my birthday party, who all arrive independently of one another. The arrival times of each person are uniformly distributed between \(07.00\ \text{pm}\) and \(08.00\ \text{pm}\). The probability that the \(13\)th person to arrive gets there between \(07.10\ \text{pm}\) and \(07.30\ \text{pm}\) is \(\large\color{blue}{p}\). Find \(\left\lfloor10^{13}\large\color{blue}{p}\right\rfloor\).

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