Exercises

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
   
   $$P(s) = A s \exp\left(-\frac{s}{s_0}\right),$$
   
   
   where $0\leq s\leq \infty$. Here, $A$ and $s_0$ are positive constants. More explicitly, $P(s) ds$ gives the probability that a car has a speed between $s$ and $s+ds$.
   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 $w$: i.e., the probability of decay in a time interval $dt$ is $w dt$. Let $P(t)$ be the probability of the atom not having decayed at time $t$, given that it was created at time $t=0$. Demonstrate that
   
   $$P(t) = {\rm e}^{-w t}.$$
   
   
   What is the mean lifetime of the atom?