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
P(s)=Asexp(−ss0)
where 0≤s≤∞. Here, A and s0 are positive constants. More explicitly, P(s)ds gives the probability that a car has a speed between s and s+ds.
- 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 w: i.e., the probability of decay in a time interval dt is wdt. Let P(t) be the probability of the atom not having decayed at time , given that it was created at time
. Demonstrate that
P(t)=e−wt
What is the mean lifetime of the atom?
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)