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2.6: Exercises

  • Page ID
    1189
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    1. In the ``game'' of Russian roulette, the player inserts a single cartridge into the drum of a revolver, leaving the other five chambers of the drum empty. The player then spins the drum, aims at his/her head, and pulls the trigger.

    1. What is the probability of the player still being alive after playing the game N times?
    2. What is the probability of the player surviving N-1 turns in this game, and then being shot the $N$th time he/she pulls the trigger?
    3. What is the mean number of times the player gets to pull the trigger?

     

    2. Suppose that the probability density for the speed $s$ of a car on a road is given by

    \begin{equation}P(s)=A s \exp \left(-\frac{s}{s_{0}}\right)\end{equation}

    where \(\begin{equation}0 \leq s \leq \infty\end{equation}\). Here, \(\begin{equation}A \text { and } s_{0}\end{equation}\) are positive constants. More explicitly, \(\begin{equation}P(s) d s\end{equation}\) gives the probability that a car has a speed between \(\begin{equation}s \text { and } s+d s\end{equation}\).

     

    1. Determine $A$ in terms of $s_0$.
    2. What is the mean value of the speed?
    3. What is the ``most probable'' speed: i.e., the speed for which the probability density has a maximum?
    4. What is the probability that a car has a speed more than three times as large as the mean value?

    3. An radioactive atom has a uniform decay probability per unit time \(\begin{equation}w\end{equation}\): i.e., the probability of decay in a time interval \(\begin{equation}d t \text { is } w d t . \text { Let } P(t)\end{equation}\) be the probability of the atom not having decayed at time $t$, given that it was created at time $t=0$. Demonstrate that

    \begin{equation}P(t)=\mathbf{e}^{-w t}\end{equation}

    What is the mean lifetime of the atom?

    Contributors

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)

    This page titled 2.6: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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