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Physics LibreTexts

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.

  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)=Asexp(ss0)

where 0s. 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.

 

  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 wi.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 $t$, given that it was created at time $t=0$. Demonstrate that

P(t)=ewt

What is the mean lifetime of the atom?

Contributors

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


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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