- Let
and
.
Show that
- Show that
- Confirm that

where - Show that the probability of throwing 6 points or less with three (six-sided) dice is
.
- Consider a game in which six (six-sided) dice are rolled. Find the probability
of obtaining:
- exactly one ace.
- at least one ace.
- exactly two aces.

- In the game of Russian roulette, one inserts a single cartridge into the drum
of a revolver, leaving the other five chambers of the drum empty. One then spins
the drum, aims at one's head, and pulls the trigger.
- Show that the probability of still being alive after playing the game times is .
- Show that the probability of surviving turns in this game, and then being shot the th times one pulls the trigger, is .
- Show that the mean number of times a player gets to pull the trigger is .

- A battery of total emf
is connected to a resistor
. As a result, an
amount of power
is dissipated in the resistor. The battery itself
consists of
individual cells connected in series, so that
is equal
to the sum of the emf's of all these cells. The battery is old, however, so that
not all cells are in perfect condition. Thus, there is a probability
that the
emf of any individual cell has its normal value
; and a probability
that the emf of any individual cell is zero because the cell has become internally
shorted. The individual cells are statistically independent of each other.
Under these conditions, show that the mean power,
, dissipated in the
resistor, is
- A drunk starts out from a lamppost in the middle of a street, taking steps
of uniform length
to the right or to the left with equal probability.
- Show that the average distance from the lamppost after steps is zero.
- Show that the root-mean-square distance (i.e. the square-root of the mean of the distance squared) from the lamppost after steps is .
- Show that the probability that the drunk will return to the lamppost after
steps is zero if
is odd, and

- A molecule in a gas moves equal distances
between collisions with equal
probabilities in any direction. Show that, after a total of
such displacements,
the mean-square displacement,
, of the molecule from its starting
point is
.
- A penny is tossed 400 times. Find the probability of getting 215 heads.
(Use the Gaussian distribution.)
- Suppose that the probability density for the speed
of a car on a highway
is given by
- Determine in terms of .
- What is the mean value of the speed?
- What is the ``most probable'' speed: that is, 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?