2.6: Exercises

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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.

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

2. Suppose that the probability density for the speed 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 $\begin{equation}s \text { and } s+d s\end{equation}$.

Determine in terms of .
What is the mean value of the speed?
What is the ``most probable'' speed: i.e., the speed for which the probability density has a maximum?
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 , given that it was created at time . 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)

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